Fix CancelledError stopping other cleanup contexts completing

[Dreamsorcerer]

I wonder if we should even just catch BaseException here...?
Then a KeyboardInterrupt or similar could cancel each context individually.

Fixes #5672.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.25%. Comparing base (0478f14) to head (5b63246).
Report is 24 commits behind head on master.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #8908      +/-   ##
==========================================
+ Coverage   98.24%   98.25%   +0.01%     
==========================================
  Files         107      107              
  Lines       34130    34126       -4     
  Branches     4050     4049       -1     
==========================================
  Hits        33531    33531              
+ Misses        423      421       -2     
+ Partials      176      174       -2     

Flag	Coverage Δ
CI-GHA	98.15% <100.00%> (+0.01%)	⬆️
OS-Linux	97.81% <100.00%> (+0.01%)	⬆️
OS-Windows	96.21% <100.00%> (+0.01%)	⬆️
OS-macOS	97.48% <100.00%> (+0.01%)	⬆️
Py-3.10.11	97.58% <100.00%> (+0.01%)	⬆️
Py-3.10.14	97.51% <100.00%> (+0.01%)	⬆️
Py-3.11.9	97.74% <100.00%> (+0.01%)	⬆️
Py-3.12.4	96.01% <100.00%> (+0.01%)	⬆️
Py-3.12.5	97.54% <100.00%> (+0.01%)	⬆️
Py-3.9.13	97.47% <100.00%> (+0.01%)	⬆️
Py-3.9.19	97.41% <100.00%> (+0.01%)	⬆️
Py-pypy7.3.16	97.02% <100.00%> (+0.01%)	⬆️
VM-macos	97.48% <100.00%> (+0.01%)	⬆️
VM-ubuntu	97.81% <100.00%> (+0.01%)	⬆️
VM-windows	96.21% <100.00%> (+0.01%)	⬆️

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[patchback]

Backport to 3.10: 💔 cherry-picking failed — conflicts found

❌ Failed to cleanly apply 032fb57 on top of patchback/backports/3.10/032fb571f2c73605d855d4f17026b303c7441823/pr-8908

Backporting merged PR #8908 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:/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/3.10/032fb571f2c73605d855d4f17026b303c7441823/pr-8908 upstream/3.10

4. Now, cherry-pick PR Fix CancelledError stopping other cleanup contexts completing #8908 contents into that branch:

   $ git cherry-pick -x 032fb571f2c73605d855d4f17026b303c7441823

   If it'll yell at you with something like fatal: Commit 032fb571f2c73605d855d4f17026b303c7441823 is a merge but no -m option was given., add -m 1 as follows instead:

   $ git cherry-pick -m1 -x 032fb571f2c73605d855d4f17026b303c7441823

5. At this point, you'll probably encounter some merge conflicts. You must
   resolve them in to preserve the patch from PR Fix CancelledError stopping other cleanup contexts completing #8908 as close to the
   original as possible.
6. Push this branch to your fork on GitHub:

   $ git push origin patchback/backports/3.10/032fb571f2c73605d855d4f17026b303c7441823/pr-8908

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!

🤖 @patchback
I'm built with octomachinery and
my source is open — https:/sanitizers/patchback-github-app.

[patchback]

Backport to 3.11: 💔 cherry-picking failed — conflicts found

❌ Failed to cleanly apply 032fb57 on top of patchback/backports/3.11/032fb571f2c73605d855d4f17026b303c7441823/pr-8908

Backporting merged PR #8908 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:/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/3.11/032fb571f2c73605d855d4f17026b303c7441823/pr-8908 upstream/3.11

4. Now, cherry-pick PR Fix CancelledError stopping other cleanup contexts completing #8908 contents into that branch:

   $ git cherry-pick -x 032fb571f2c73605d855d4f17026b303c7441823

   If it'll yell at you with something like fatal: Commit 032fb571f2c73605d855d4f17026b303c7441823 is a merge but no -m option was given., add -m 1 as follows instead:

   $ git cherry-pick -m1 -x 032fb571f2c73605d855d4f17026b303c7441823

5. At this point, you'll probably encounter some merge conflicts. You must
   resolve them in to preserve the patch from PR Fix CancelledError stopping other cleanup contexts completing #8908 as close to the
   original as possible.
6. Push this branch to your fork on GitHub:

   $ git push origin patchback/backports/3.11/032fb571f2c73605d855d4f17026b303c7441823/pr-8908

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!

🤖 @patchback
I'm built with octomachinery and
my source is open — https:/sanitizers/patchback-github-app.