[PR #8457/db1b9129 backport][3.10] Use async fixture in testing docs

[patchback]

This is a backport of PR #8457 as merged into master (db1b912).

Instead of low-level asyncio APIs, which regular users shouldn't be using.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

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Additional details and impacted files

@@           Coverage Diff           @@
##             3.10    #8461   +/-   ##
=======================================
  Coverage        ?   97.51%           
=======================================
  Files           ?      108           
  Lines           ?    33282           
  Branches        ?     3994           
=======================================
  Hits            ?    32456           
  Misses          ?      602           
  Partials        ?      224           

Flag	Coverage Δ
CI-GHA	97.42% <ø> (?)
OS-Linux	97.10% <ø> (?)
OS-Windows	94.70% <ø> (?)
OS-macOS	96.76% <ø> (?)
Py-3.10.11	96.87% <ø> (?)
Py-3.10.14	96.84% <ø> (?)
Py-3.11.9	97.06% <ø> (?)
Py-3.12.3	97.18% <ø> (?)
Py-3.8.10	94.44% <ø> (?)
Py-3.8.18	96.73% <ø> (?)
Py-3.9.13	96.87% <ø> (?)
Py-3.9.19	96.84% <ø> (?)
Py-pypy7.3.16	96.40% <ø> (?)
VM-macos	96.76% <ø> (?)
VM-ubuntu	97.10% <ø> (?)
VM-windows	94.70% <ø> (?)

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