[PR #8271/e0d9d3ab backport][3.9] Fix race condition with request handlers on shutdown

[patchback]

This is a backport of PR #8271 as merged into master (e0d9d3a).

Fixes #8212.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.42%. Comparing base (3a13bd5) to head (651f970).

Additional details and impacted files

@@            Coverage Diff             @@
##              3.9    #8272      +/-   ##
==========================================
+ Coverage   97.40%   97.42%   +0.02%     
==========================================
  Files         108      108              
  Lines       32796    32797       +1     
  Branches     3876     3876              
==========================================
+ Hits        31945    31953       +8     
+ Misses        647      640       -7     
  Partials      204      204              

Flag	Coverage Δ
CI-GHA	97.33% <100.00%> (+0.02%)	⬆️
OS-Linux	97.01% <100.00%> (+<0.01%)	⬆️
OS-Windows	94.61% <100.00%> (+0.02%)	⬆️
OS-macOS	96.73% <100.00%> (+0.01%)	⬆️
Py-3.10.11	94.42% <100.00%> (+0.02%)	⬆️
Py-3.10.13	96.54% <100.00%> (+0.01%)	⬆️
Py-3.10.14	96.76% <100.00%> (+0.01%)	⬆️
Py-3.11.8	97.04% <100.00%> (+0.02%)	⬆️
Py-3.12.2	97.10% <100.00%> (+0.02%)	⬆️
Py-3.8.10	94.40% <100.00%> (+0.01%)	⬆️
Py-3.8.18	96.73% <100.00%> (+0.01%)	⬆️
Py-3.9.13	94.41% <100.00%> (+0.02%)	⬆️
Py-3.9.18	96.52% <100.00%> (+0.01%)	⬆️
Py-3.9.19	96.75% <100.00%> (+0.01%)	⬆️
Py-pypy7.3.15	96.32% <100.00%> (+<0.01%)	⬆️
VM-macos	96.73% <100.00%> (+0.01%)	⬆️
VM-ubuntu	97.01% <100.00%> (+<0.01%)	⬆️
VM-windows	94.61% <100.00%> (+0.02%)	⬆️

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