fix readthedocs (RTD) public docsite, modernize some of the docs build/testing

[briantist]

Fixes #1061
Resolves #1053
Resolves #1076
Closes #1109

I got added to the RTD maintainers and was able to update several settings, like our defualt branch.
I've also enabled pull request builds, so we can see when a PR would break RTD (or in the case of this PR, fixes it), and maintainers can also view the generated documentation which is nice.

Fixing the build involved updating a lot of outdated requirements, so I also moved all the requirements into poetry as described in #1053 .

Along the way, updated the extremely our-of-date sphinx and sphinx-rtd-theme versions, so maybe the docs will even look a little nicer ✨.

The requirements upgrades needed some changes throughout other test code to remove some deprecated things.

I didn't go as far as trying to untangle our custom doctest thing and remove that, there's enough in this PR already and I want to get the public docsite fixed. That work will probably be part of or a pre-requisite for trying to move doctests into pytest, described a bit in #1105 .

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (7db7de1) 87.12% compared to head (e0583c6) 87.12%.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #1119   +/-   ##
=======================================
  Coverage   87.12%   87.12%           
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  Files          64       64           
  Lines        3162     3162           
=======================================
  Hits         2755     2755           
  Misses        407      407           

[briantist]

All changes here only affect dev/tests,a nd the public docsite, no changes to the library.