fix: Fix dangling coroutines in request extraction handling cleanup #4003

Job	Run time
mypy	1m 15s
slotscheck	31s
pyright	58s
Test server integration (uvicorn<0.27.0)	1m 4s
Test Minimal Application with Base Dependencies	34s
Test Minimal Pydantic 1 application	35s
Test server integration (uvicorn>=0.27.0)	55s
validate	12s
test-platform-compat	0s
test (3.10) / test	1m 57s
test (3.12) / test	3m 26s
test (3.11) / test	1m 52s
test (3.8) / test	3m 3s
test (3.9) / test	2m 0s
build-docs	2m 41s
codeql	3m 5s
upload-test-coverage	15s
sonar	1m 9s
	25m 32s

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