One of the underlying ideas in gauge invariance is that the physics exist independently of whichever coordinate parameterization (and thus gauge) we apply to the system.

For unit-preserving changes in coordinates, the Lagrangian (as a measure of the "energy" in the system") should thus have a value independent of which coordinates we choose.

One of the underlying ideas in gauge invariance is that the physics exist independently of the coordinate parameterization (and thus gauge) we apply to the system.

For unit-preserving changes in coordinates, the Lagrangian (as a measure of the "energy" in the system") should thus have a value independent of which coordinates we choose.

The motivation for covariant derivatives, etc., is then to account for disagreements between coordinate systems (and the underlying physical manifold) of what makes a "straight line" or a "right angle".