Simplify keep-alive logic for HTTP/1.0 in ClientRequest

[bdraco]

Not a functional change. Noticed while looking for another issue that this code could be made more concise

[codspeed-hq]

CodSpeed Performance Report

Merging #9704 will not alter performance

_{Comparing client_reqrep_keepalive_cleanup (c65f861) with master (dd0b6e3)}

Summary

✅ 14 untouched benchmarks

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.67%. Comparing base (dd0b6e3) to head (c65f861).
Report is 2 commits behind head on master.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #9704      +/-   ##
==========================================
- Coverage   98.67%   98.67%   -0.01%     
==========================================
  Files         116      116              
  Lines       35733    35731       -2     
  Branches     4238     4237       -1     
==========================================
- Hits        35259    35257       -2     
  Misses        319      319              
  Partials      155      155              

Flag	Coverage Δ
CI-GHA	98.55% <100.00%> (-0.01%)	⬇️
OS-Linux	98.23% <100.00%> (-0.01%)	⬇️
OS-Windows	96.12% <100.00%> (-0.01%)	⬇️
OS-macOS	97.43% <100.00%> (+<0.01%)	⬆️
Py-3.10.11	97.28% <100.00%> (-0.01%)	⬇️
Py-3.10.15	97.71% <100.00%> (-0.01%)	⬇️
Py-3.11.10	97.77% <100.00%> (-0.01%)	⬇️
Py-3.11.9	97.35% <100.00%> (+<0.01%)	⬆️
Py-3.12.7	98.26% <100.00%> (-0.01%)	⬇️
Py-3.13.0	98.24% <100.00%> (-0.01%)	⬇️
Py-3.9.13	97.20% <100.00%> (-0.01%)	⬇️
Py-3.9.20	97.63% <100.00%> (-0.01%)	⬇️
Py-pypy7.3.16	97.25% <100.00%> (-0.01%)	⬇️
VM-macos	97.43% <100.00%> (+<0.01%)	⬆️
VM-ubuntu	98.23% <100.00%> (-0.01%)	⬇️
VM-windows	96.12% <100.00%> (-0.01%)	⬇️

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[patchback]

Backport to 3.10: 💚 backport PR created

✅ Backport PR branch: patchback/backports/3.10/26d96107d84d8e5c30ae38b86456634b2e210cf2/pr-9704

Backported as #9706

🤖 @patchback
I'm built with octomachinery and
my source is open — https://github.com/sanitizers/patchback-github-app.

[patchback]

Backport to 3.11: 💚 backport PR created

✅ Backport PR branch: patchback/backports/3.11/26d96107d84d8e5c30ae38b86456634b2e210cf2/pr-9704

Backported as #9707

🤖 @patchback
I'm built with octomachinery and
my source is open — https://github.com/sanitizers/patchback-github-app.