[PR #8309/c29945a1 backport][3.9] Improve reliability of run_app test

[patchback]

This is a backport of PR #8309 as merged into master (c29945a).

Have seen a flaky test failure where the server failed to accept the request in time, on mac.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.39%. Comparing base (ec2be05) to head (2b21632).

Additional details and impacted files

@@            Coverage Diff             @@
##              3.9    #8315      +/-   ##
==========================================
- Coverage   97.39%   97.39%   -0.01%     
==========================================
  Files         108      108              
  Lines       32819    32817       -2     
  Branches     3891     3889       -2     
==========================================
- Hits        31964    31962       -2     
  Misses        645      645              
  Partials      210      210              

Flag	Coverage Δ
CI-GHA	97.30% <100.00%> (-0.01%)	⬇️
OS-Linux	96.98% <100.00%> (-0.01%)	⬇️
OS-Windows	94.51% <100.00%> (-0.07%)	⬇️
OS-macOS	96.69% <100.00%> (-0.01%)	⬇️
Py-3.10.11	94.39% <100.00%> (-0.01%)	⬇️
Py-3.10.14	96.73% <100.00%> (-0.03%)	⬇️
Py-3.11.8	96.89% <100.00%> (-0.14%)	⬇️
Py-3.11.9	96.47% <100.00%> (?)
Py-3.12.2	97.04% <100.00%> (-0.03%)	⬇️
Py-3.8.10	94.37% <100.00%> (-0.01%)	⬇️
Py-3.8.18	96.70% <100.00%> (-0.01%)	⬇️
Py-3.9.13	?
Py-3.9.19	96.74% <100.00%> (-0.01%)	⬇️
Py-pypy7.3.15	96.29% <100.00%> (-0.01%)	⬇️
VM-macos	96.69% <100.00%> (-0.01%)	⬇️
VM-ubuntu	96.98% <100.00%> (-0.01%)	⬇️
VM-windows	94.51% <100.00%> (-0.07%)	⬇️

Flags with carried forward coverage won't be shown. Click here to find out more.

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.