Cherry-pick some pytest-asyncio fixes

[Dreamsorcerer]

Some of the fixes from #10762, in order to reduce maintenance burden while that is still ongoing.
These include some general improvements and nothing actually depends on pytest-asyncio.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.84%. Comparing base (2dcf534) to head (d89bde7).
⚠️ Report is 212 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master   #11247      +/-   ##
==========================================
- Coverage   98.84%   98.84%   -0.01%     
==========================================
  Files         131      131              
  Lines       43280    43249      -31     
  Branches     2328     2326       -2     
==========================================
- Hits        42782    42748      -34     
- Misses        343      346       +3     
  Partials      155      155              

Flag	Coverage Δ
CI-GHA	98.73% <100.00%> (-0.01%)	⬇️
OS-Linux	98.46% <100.00%> (-0.01%)	⬇️
OS-Windows	96.79% <100.00%> (-0.01%)	⬇️
OS-macOS	97.63% <100.00%> (-0.05%)	⬇️
Py-3.10.11	97.33% <100.00%> (-0.01%)	⬇️
Py-3.10.18	97.81% <100.00%> (-0.01%)	⬇️
Py-3.11.13	98.00% <100.00%> (-0.01%)	⬇️
Py-3.11.9	97.53% <100.00%> (-0.01%)	⬇️
Py-3.12.10	96.47% <100.00%> (-1.16%)	⬇️
Py-3.12.11	98.10% <100.00%> (-0.01%)	⬇️
Py-3.13.3	98.36% <100.00%> (-0.01%)	⬇️
Py-3.9.13	96.05% <100.00%> (-1.18%)	⬇️
Py-3.9.23	97.69% <100.00%> (-0.01%)	⬇️
Py-pypy7.3.16	86.78% <100.00%> (-7.88%)	⬇️
VM-macos	97.63% <100.00%> (-0.05%)	⬇️
VM-ubuntu	98.46% <100.00%> (-0.01%)	⬇️
VM-windows	96.79% <100.00%> (-0.01%)	⬇️

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[codspeed-hq]

CodSpeed Performance Report

Merging #11247 will not alter performance

_{Comparing Dreamsorcerer-patch-5 (d89bde7) with master (2dcf534)}

Summary

✅ 59 untouched benchmarks

[patchback]

Backport to 3.13: 💔 cherry-picking failed — conflicts found

❌ Failed to cleanly apply 55e4300 on top of patchback/backports/3.13/55e430051259171f249fba97d080abef04c15717/pr-11247

Backporting merged PR #11247 into master

1. Ensure you have a local repo clone of your fork. Unless you cloned it
   from the upstream, this would be your origin remote.
2. Make sure you have an upstream repo added as a remote too. In these
   instructions you'll refer to it by the name upstream. If you don't
   have it, here's how you can add it:

   $ git remote add upstream https://github.com/aio-libs/aiohttp.git

3. Ensure you have the latest copy of upstream and prepare a branch
   that will hold the backported code:

   $ git fetch upstream
   $ git checkout -b patchback/backports/3.13/55e430051259171f249fba97d080abef04c15717/pr-11247 upstream/3.13

4. Now, cherry-pick PR Cherry-pick some pytest-asyncio fixes #11247 contents into that branch:

   $ git cherry-pick -x 55e430051259171f249fba97d080abef04c15717

   If it'll yell at you with something like fatal: Commit 55e430051259171f249fba97d080abef04c15717 is a merge but no -m option was given., add -m 1 as follows instead:

   $ git cherry-pick -m1 -x 55e430051259171f249fba97d080abef04c15717

5. At this point, you'll probably encounter some merge conflicts. You must
   resolve them in to preserve the patch from PR Cherry-pick some pytest-asyncio fixes #11247 as close to the
   original as possible.
6. Push this branch to your fork on GitHub:

   $ git push origin patchback/backports/3.13/55e430051259171f249fba97d080abef04c15717/pr-11247

7. Create a PR, ensure that the CI is green. If it's not — update it so that
   the tests and any other checks pass. This is it!
   Now relax and wait for the maintainers to process your pull request
   when they have some cycles to do reviews. Don't worry — they'll tell you if
   any improvements are necessary when the time comes!

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