Update Sphinx docs to cover the API.

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This is probably one of the most important issues right now that the current docs are pretty terrible and people are usually forced to use the old docs or go into the headers for any new API features.

I'm making this issue because I keep forgetting the steps to do this so I need to organize them here.

The main issue is that I'm using the wrong directives most of the time. I'm using the doxygenclass and doxygenfunction directives a lot which makes it really hard to keep the docs up-to-date with the headers, especially when overloads are used. I need to be using the doxygenfile or autodoxygenfile directives instead to document entire header files automatically and only use the more specific directives for special cases.
https://breathe.readthedocs.io/en/latest/directives.html

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doxygengroup seems to be the ideal directive for the older docs. This lets me group things nicely without needing to change the overall structure.

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Breathe directives are causing conflicts, so I'm looking into using Exhale to extract the docs from the sources instead.

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So far all docs that were only in the headers are now in the online docs. However it's mostly auto-generated and there are multiple symbols included which should be removed.

I can't figure out how to effectively use Doxygen groups. In theory this should let me group related functions together on one page of docs but I'm not sure how it works in practice with Breathe/Exhale.