[PR #8495/549c95b9 backport][3.10] Shutdown logic: Only wait on handlers

[bdraco]

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: J. Nick Koston [email protected]
(cherry picked from commit 549c95b)

[codecov]

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@@           Coverage Diff           @@
##             3.10    #8530   +/-   ##
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  Coverage        ?   97.56%           
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  Files           ?      108           
  Lines           ?    33518           
  Branches        ?     4029           
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  Hits            ?    32702           
  Misses          ?      594           
  Partials        ?      222           

Flag	Coverage Δ
CI-GHA	97.47% <100.00%> (?)
OS-Linux	97.14% <100.00%> (?)
OS-Windows	94.73% <100.00%> (?)
OS-macOS	96.81% <100.00%> (?)
Py-3.10.11	96.92% <100.00%> (?)
Py-3.10.14	96.87% <100.00%> (?)
Py-3.11.9	97.10% <100.00%> (?)
Py-3.12.4	97.22% <100.00%> (?)
Py-3.8.10	94.47% <100.00%> (?)
Py-3.8.18	96.76% <100.00%> (?)
Py-3.9.13	96.91% <100.00%> (?)
Py-3.9.19	96.88% <100.00%> (?)
Py-pypy7.3.16	96.44% <100.00%> (?)
VM-macos	96.81% <100.00%> (?)
VM-ubuntu	97.14% <100.00%> (?)
VM-windows	94.73% <100.00%> (?)

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