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 (597161d) to head (ad0799c).
Report is 2 commits behind head on master.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##           master   #10941   +/-   ##
=======================================
  Coverage   98.76%   98.77%           
=======================================
  Files         129      129           
  Lines       39455    39458    +3     
  Branches     2184     2185    +1     
=======================================
+ Hits        38967    38973    +6     
+ Misses        340      337    -3     
  Partials      148      148           

Flag	Coverage Δ
CI-GHA	98.65% <66.66%> (+<0.01%)	⬆️
OS-Linux	98.36% <66.66%> (+<0.01%)	⬆️
OS-Windows	96.53% <66.66%> (-0.01%)	⬇️
OS-macOS	97.51% <66.66%> (-0.01%)	⬇️
Py-3.10.11	97.41% <33.33%> (+<0.01%)	⬆️
Py-3.10.17	97.93% <33.33%> (-0.01%)	⬇️
Py-3.11.12	98.01% <33.33%> (-0.01%)	⬇️
Py-3.11.9	97.48% <33.33%> (-0.02%)	⬇️
Py-3.12.10	98.42% <33.33%> (+<0.01%)	⬆️
Py-3.13.3	98.41% <33.33%> (-0.01%)	⬇️
Py-3.9.13	97.27% <33.33%> (-0.01%)	⬇️
Py-3.9.22	97.80% <33.33%> (-0.01%)	⬇️
Py-pypy7.3.16	84.00% <33.33%> (-8.89%)	⬇️
VM-macos	97.51% <66.66%> (-0.01%)	⬇️
VM-ubuntu	98.36% <66.66%> (+<0.01%)	⬆️
VM-windows	96.53% <66.66%> (-0.01%)	⬇️

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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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