Use anext in AsyncIterablePayload on Python 3.10+

[bdraco]

Implements #10915 (comment)

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.77%. Comparing base (38c23ed) to head (ad0799c).
⚠️ Report is 3145 commits behind head on master.
✅ All tests successful. No failed tests found.

Additional details and impacted files

@@            Coverage Diff             @@
##           master   #10941      +/-   ##
==========================================
+ Coverage   98.16%   98.77%   +0.60%     
==========================================
  Files         130      129       -1     
  Lines       39652    39458     -194     
  Branches     2240     2185      -55     
==========================================
+ Hits        38925    38973      +48     
+ Misses        555      337     -218     
+ Partials      172      148      -24     

Flag	Coverage Δ
CI-GHA	98.65% <66.66%> (+0.59%)	⬆️
OS-Linux	98.36% <66.66%> (+0.57%)	⬆️
OS-Windows	96.53% <66.66%> (+1.27%)	⬆️
OS-macOS	97.51% <66.66%> (+0.55%)	⬆️
Py-3.10.11	97.41% <33.33%> (+0.56%)	⬆️
Py-3.10.17	97.93% <33.33%> (+0.55%)	⬆️
Py-3.11.12	98.01% <33.33%> (+0.53%)	⬆️
Py-3.11.9	97.48% <33.33%> (+0.54%)	⬆️
Py-3.12.10	98.42% <33.33%> (+0.58%)	⬆️
Py-3.13.3	98.41% <33.33%> (+0.57%)	⬆️
Py-3.9.13	97.27% <33.33%> (+0.56%)	⬆️
Py-3.9.22	97.80% <33.33%> (+0.53%)	⬆️
Py-pypy7.3.16	84.00% <33.33%> (-12.82%)	⬇️
VM-macos	97.51% <66.66%> (+0.55%)	⬆️
VM-ubuntu	98.36% <66.66%> (+0.57%)	⬆️
VM-windows	96.53% <66.66%> (+1.27%)	⬆️

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[codspeed-hq]

CodSpeed Performance Report

Merging #10941 will not alter performance

_{Comparing anext_payload (ad0799c) with master (77c0115)}

Summary

✅ 59 untouched benchmarks

[patchback]

Backport to 3.12: 💚 backport PR created

✅ Backport PR branch: patchback/backports/3.12/6512aaa436b7252d5206a1a3a5b5feae4ef2b829/pr-10941

Backported as #10942

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