[PR #8681/30a3d0ef backport][3.10] Improve performance of starting request handlers with Python 3.12+

[patchback]

This is a backport of PR #8681 as merged into master (30a3d0e).

What do these changes do?

Improve performance of starting request handlers with Python 3.12+ by starting the task eagerly.

followup to #8661 as I missed one more place we create a task

Are there changes in behavior for the user?

no

Is it a substantial burden for the maintainers to support this?

no

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.69%. Comparing base (78d45e7) to head (8164e70).
Report is 5 commits behind head on 3.10.

Additional details and impacted files

@@            Coverage Diff             @@
##             3.10    #8725      +/-   ##
==========================================
+ Coverage   96.71%   97.69%   +0.97%     
==========================================
  Files         106      108       +2     
  Lines       34018    34092      +74     
  Branches     3264     4058     +794     
==========================================
+ Hits        32901    33305     +404     
+ Misses        854      587     -267     
+ Partials      263      200      -63     

Flag	Coverage Δ
CI-GHA	97.58% <83.33%> (+0.86%)	⬆️
OS-Linux	97.26% <83.33%> (?)
OS-Windows	94.79% <83.33%> (?)
OS-macOS	96.92% <83.33%> (+0.21%)	⬆️
Py-3.10.11	97.02% <66.66%> (+0.30%)	⬆️
Py-3.10.14	96.97% <66.66%> (?)
Py-3.11.9	97.20% <66.66%> (?)
Py-3.12.4	97.30% <66.66%> (?)
Py-3.8.10	94.40% <66.66%> (?)
Py-3.8.18	96.75% <66.66%> (?)
Py-3.9.13	96.90% <66.66%> (?)
Py-3.9.19	96.85% <66.66%> (?)
Py-pypy7.3.16	96.45% <66.66%> (?)
VM-macos	96.92% <83.33%> (+0.21%)	⬆️
VM-ubuntu	97.26% <83.33%> (?)
VM-windows	94.79% <83.33%> (?)

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