fix: Fix dangling coroutines in request extraction handling cleanup #4013

Job	Run time
validate	21s
mypy	1m 15s
pyright	51s
slotscheck	29s
Test server integration (uvicorn<0.27.0)	50s
Test server integration (uvicorn>=0.27.0)	58s
Test Minimal Application with Base Dependencies	31s
Test Minimal Pydantic 1 application	39s
test-platform-compat	0s
test (3.10) / test	2m 5s
test (3.11) / test	1m 54s
test (3.12) / test	3m 35s
test (3.8) / test	2m 57s
test (3.9) / test	2m 0s
build-docs	3m 8s
codeql	3m 9s
upload-test-coverage	19s
sonar	1m 21s
	26m 22s

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