[OP] According to this page along with many others, oxidation and reduction must occur simultaneously. However, in a battery, it seems like they actually occur separately

Simultaneously means at the same time. Separately could mean independently (not the case here) or in different locations (that's the appropriate interpretation).

[OP] Oxidation must first occur to provide the electrons for reduction, and since these electrons cannot pass through the electrolyte barrier they have to go through the circuit, which takes time. This not only staggers the oxidation and reduction reactions, but it also isolates the electrons being transferred, which should be impossible if reduction and oxidation can only happen together.

This is an inaccurate description. The electrons don't have to travel all the way, the wire is filled with electrons already (using a classical view of electrons). On a particular level, oxidation and reduction can occur independently (in fact, if you just have a half-cell, there will be oxidation and reduction events, but they don't result in a net reaction). At a bulk level, they are coordinated and occur virtually simultaneously.

Oxidation or reduction can occur "first". For reduction, extra electrons can go into the electrode and circuit. For oxidation, needed electrons can be taken from the electrode and circuit. Both just to a certain point because the charge buildup encourages the reverse reaction.

[OP] They are not only physically separated but also it is as if each reaction is its own half-reaction, with electrons being isolated as a product and reactant.

The electrons are the key here. You can't run bulk half reactions because there is no source or sink for the electrons unless you pair oxidation half reaction with reduction half reaction. It's like trying to sell your cup cakes at a bake sale when there are no buyers, or trying to eat cup cakes when there is no one to bake them.

[OP] According to this page along with many others, oxidation and reduction must occur simultaneously. However, in a battery, it seems like they actually occur separately

Simultaneously means at the same time. Separately could mean independently (not the case here) or in different locations (that's the appropriate interpretation).

[OP] Oxidation must first occur to provide the electrons for reduction, and since these electrons cannot pass through the electrolyte barrier they have to go through the circuit, which takes time. This not only staggers the oxidation and reduction reactions, but it also isolates the electrons being transferred, which should be impossible if reduction and oxidation can only happen together.

This is an inaccurate description. The electrons don't have to travel all the way, the wire is filled with electrons already (using a classical view of electrons). On a particular level, oxidation and reduction can occur independently (in fact, if you just have a half-cell, there will be oxidation and reduction events, but they don't result in a net reaction). At a bulk level, they are coordinated and occur virtually simultaneously.

Oxidation or reduction can occur "first". For oxidation, extra electrons can go into the electrode and circuit. For reduction, needed electrons can be taken from the electrode and circuit. Both just react to a certain point because the charge buildup encourages the reverse reactions.

[OP] They are not only physically separated but also it is as if each reaction is its own half-reaction, with electrons being isolated as a product and reactant.

The electrons are the key here. You can't run bulk half reactions because there is no source or sink for the electrons unless you pair oxidation half reaction with reduction half reaction. It's like trying to sell your cup cakes at a bake sale when there are no buyers, or trying to eat cup cakes when there is no one to bake them.