[PR #9340/8a97e03 backport][3.10] Use dunder writer internally in ClientResponse

[bdraco]

(cherry picked from commit 8a97e03)

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.78%. Comparing base (b959358) to head (d372917).
Report is 109 commits behind head on 3.10.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@            Coverage Diff             @@
##             3.10    #9341      +/-   ##
==========================================
- Coverage   97.79%   97.78%   -0.01%     
==========================================
  Files         108      108              
  Lines       34885    34883       -2     
  Branches     4173     4173              
==========================================
- Hits        34115    34111       -4     
- Misses        573      574       +1     
- Partials      197      198       +1     

Flag	Coverage Δ
CI-GHA	97.67% <100.00%> (-0.01%)	⬇️
OS-Linux	97.35% <100.00%> (-0.01%)	⬇️
OS-Windows	94.97% <100.00%> (-0.01%)	⬇️
OS-macOS	97.06% <100.00%> (+<0.01%)	⬆️
Py-3.10.11	96.92% <100.00%> (-0.01%)	⬇️
Py-3.10.15	96.86% <100.00%> (-0.01%)	⬇️
Py-3.11.10	96.94% <100.00%> (-0.01%)	⬇️
Py-3.11.9	97.00% <100.00%> (-0.03%)	⬇️
Py-3.12.6	97.40% <100.00%> (-0.01%)	⬇️
Py-3.13.0-rc.2	97.38% <100.00%> (-0.01%)	⬇️
Py-3.8.10	94.36% <100.00%> (-0.01%)	⬇️
Py-3.8.18	96.65% <100.00%> (-0.01%)	⬇️
Py-3.9.13	96.83% <100.00%> (-0.01%)	⬇️
Py-3.9.20	96.78% <100.00%> (-0.01%)	⬇️
Py-pypy7.3.16	96.38% <100.00%> (-0.01%)	⬇️
VM-macos	97.06% <100.00%> (+<0.01%)	⬆️
VM-ubuntu	97.35% <100.00%> (-0.01%)	⬇️
VM-windows	94.97% <100.00%> (-0.01%)	⬇️

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