fix: Fix dangling coroutines in request extraction handling cleanup #4009

Job	Run time
pyright	55s
slotscheck	34s
mypy	1m 11s
validate	15s
Test server integration (uvicorn<0.27.0)	53s
Test Minimal Application with Base Dependencies	28s
Test server integration (uvicorn>=0.27.0)	57s
Test Minimal Pydantic 1 application	36s
test-platform-compat	0s
test (3.10) / test	1m 54s
test (3.11) / test	1m 59s
test (3.12) / test	3m 30s
test (3.8) / test	2m 57s
test (3.9) / test	2m 1s
build-docs	3m 5s
upload-test-coverage	16s
sonar	1m 7s
codeql	3m 24s
	26m 2s

Provide feedback