You are misunderstanding what a gauge theory is if you think we shouldn't get rid of the gauge symmetry at some point. A gauge symmetry is not like other symmetries, it does not relate configurations of the dynamical variables that are physically distinct - instead, it relates configuration of the dynamical variables which are physically indistinguishable. There is no detectable difference between any configuration and its gauge-transformed version at all. Unlike, say, a rotational symmetry where a vector pointing in one direction is distinct from its rotated version, in this case, there is really no physically meaningful distinction between configurations related by gauge symmetries. See also, for instance, this question, this question, this question and more.

Gauge symmetries reflect redundancy in the variables we have chosen to describe the system, they are entirely features of a particular theoretical choice and not inherent properties of the physical system under consideration, like e.g. rotational symmetry. There is therefore no need to try to preserve this symmetry - if it gets lost in an equivalent but more convenient description of the system, we shouldn't hesitate. It is a curious fact that rather often the gauge theoretical description turns out to be the most convenient.

Except, of course, when we want to do things like the path integral. To take the naive path integral over an action with gauge symmetry that has not been fixed is manifestly absurd physically: You are integrating over a space of dynamical variables, where each configuration of them has infinitely many different configurations that describe the exact same state of the exact same physical system, and you're integrating over all of them. What is this supposed to be? It's certainly not the integral over all possible physical paths, it's massively overcounting them and you have no way to control the manner in which it does that.

The natural physical path integral is one integrating over each physically distinct configuration once. When we completely fix a gauge, this is exactly what the gauge fixing does: From all the possible equivalent configurations, the gauge condition picks one and only one representant, and we then wish to integrate over this space of representants, as it is the space of physically distinct configurations. Unfortunately, Gribov ambiguities mean that we can usually not do that throughout all of field configurations space and may be stuck defining the path integral only over a subset of physical configurations, a so-called Gribov region.

Therefore, it is unreasonable to expect there to be a path integral without fixing a gauge. The path integral, by its very purpose, must integrate over the space of all physically distinct configurations, and the way to achieve that in a gauge theory is some manner of gauge fixing, there is no way to evade this fact.

You are misunderstanding what a gauge theory is if you think we shouldn't get rid of the gauge symmetry at some point. A gauge symmetry is not like other symmetries, it does not relate configurations of the dynamical variables that are physically distinct - instead, it relates configuration of the dynamical variables which are physically indistinguishable. There is no detectable difference between any configuration and its gauge-transformed version at all. Unlike, say, a rotational symmetry where a vector pointing in one direction is distinct from its rotated version, in this case, there is really no physically meaningful distinction between configurations related by gauge symmetries. See also, for instance, this question, this question, this question and more.

Gauge symmetries reflect redundancy in the variables we have chosen to describe the system, they are entirely features of a particular theoretical choice and not inherent properties of the physical system under consideration, like e.g. rotational symmetry. There is therefore no need to try to preserve this symmetry - if it gets lost in an equivalent but more convenient description of the system, we shouldn't hesitate. It is a curious fact that rather often the gauge theoretical description turns out to be the most convenient.

Except, of course, when we want to do things like the path integral. To take the naive path integral over an action with gauge symmetry that has not been fixed is manifestly absurd physically: You are integrating over a space of dynamical variables, where each configuration of them has infinitely many different configurations that describe the exact same state of the exact same physical system, and you're integrating over all of them. What is this supposed to be? It's certainly not the integral over all possible physical paths, it's massively overcounting them and you have no way to control the manner in which it does that.

The natural physical path integral is one integrating over each physically distinct configuration once. When we completely fix a gauge, this is exactly what the gauge fixing does: From all the possible equivalent configurations, the gauge condition picks one and only one representant, and we then wish to integrate over this space of representants, as it is the space of physically distinct configurations. Unfortunately, Gribov ambiguities mean that we can usually not do that throughout all of field configurations space and may be stuck defining the path integral only over a subset of physical configurations, a so-called Gribov region.

Therefore, it is unreasonable to expect there to be a path integral without fixing a gauge. The path integral, by its very purpose, must integrate over the space of all physically distinct configurations, and the way to achieve that in a gauge theory is some manner of gauge fixing, there is no way to evade this fact.

You are misunderstanding what a gauge theory is if you think we shouldn't get rid of the gauge symmetry at some point. A gauge symmetry is not like other symmetries, it does not relate configurations of the dynamical variables that are physically distinct - instead, it relates configuration of the dynamical variables which are physically indistinguishable. There is no detectable difference between any configuration and its gauge-transformed version at all. Unlike, say, a rotational symmetry where a vector pointing in one direction is distinct from its rotated version, in this case, there is really no physically meaningful distinction between configurations related by gauge symmetries. See also, for instance, this question, this question, this question and more.

Gauge symmetries reflect redundancy in the variables we have chosen to describe the system, they are entirely features of a particular theoretical choice and not inherent properties of the physical system under consideration, like e.g. rotational symmetry. There is therefore no need to try to preserve this symmetry - if it gets lost in an equivalent but more convenient description of the system, we shouldn't hesitate. It is a curious fact that rather often the gauge theoretical description turns out to be the most convenient.

Except, of course, when we want to do things like the path integral. To take the naive path integral over an action with gauge symmetry that has not been fixed is manifestly absurd physically: You are integrating over a space of dynamical variables, where each configuration of them has infinitely many different configurations that describe the exact same state of the exact same physical system, and you're integrating over all of them. What is this supposed to be? It's certainly not the integral over all possible physical paths, it's massively overcounting them and you have no way to control the manner in which it does that.

The natural physical path integral is one integrating over each physically distinct configuration once. When we completely fix a gauge, this is exactly what the gauge fixing does: From all the possible equivalent configurations, the gauge condition picks one and only one representant, and we then wish to integrate over this space of representants, as it is the spoace of physically distinct configurations. Unfortunately, Gribov ambiguities mean that we can usually not do that throughout all of field configurations space and may be stuck defining the path integral only over a subset of physical configurations, a so-called Gribov region.

Therefore, it is unreasonable to expect there to be a path integral without fixing a gauge. The path integral, by its very purpose, must integrate over the space of all physically distinct configurations, and the way to achieve that in a gauge theory is some manner of gauge fixing, there is no way to evade this fact.

You are misunderstanding what a gauge theory is if you think we shouldn't get rid of the gauge symmetry at some point. A gauge symmetry is not like other symmetries, it does not relate configurations of the dynamical variables that are physically distinct - instead, it relates configuration of the dynamical variables which are physically indistinguishable. There is no detectable difference between any configuration and its gauge-transformed version at all. Unlike, say, a rotational symmetry where a vector pointing in one direction is distinct from its rotated version, in this case, there is really no physically meaningful distinction between configurations related by gauge symmetries. See also, for instance, this question, this question, this question and more.

Gauge symmetries reflect redundancy in the variables we have chosen to describe the system, they are entirely features of a particular theoretical choice and not inherent properties of the physical system under consideration, like e.g. rotational symmetry. There is therefore no need to try to preserve this symmetry - if it gets lost in an equivalent but more convenient description of the system, we shouldn't hesitate. It is a curious fact that rather often the gauge theoretical description turns out to be the most convenient.

Except, of course, when we want to do things like the path integral. To take the naive path integral over an action with gauge symmetry that has not been fixed is manifestly absurd physically: You are integrating over a space of dynamical variables, where each configuration of them has infinitely many different configurations that describe the exact same state of the exact same physical system, and you're integrating over all of them. What is this supposed to be? It's certainly not the integral over all possible physical paths, it's massively overcounting them and you have no way to control the manner in which it does that.

The natural physical path integral is one integrating over each physically distinct configuration once. When we completely fix a gauge, this is exactly what the gauge fixing does: From all the possible equivalent configurations, the gauge condition picks one and only one representant, and we then wish to integrate over this space of representants, as it is the space of physically distinct configurations. Unfortunately, Gribov ambiguities mean that we can usually not do that throughout all of field configurations space and may be stuck defining the path integral only over a subset of physical configurations, a so-called Gribov region.

Therefore, it is unreasonable to expect there to be a path integral without fixing a gauge. The path integral, by its very purpose, must integrate over the space of all physically distinct configurations, and the way to achieve that in a gauge theory is some manner of gauge fixing, there is no way to evade this fact.