Deduplicate awaitable check in request handlers #3984

Job	Run time
check_release	1m 42s
Test Lint	47s
test_docs	48s
Test Minimum Versions	2m 42s
Test Prereleases	2m 35s
Make SDist	29s
test_without_debugpy (ubuntu-latest, 3.10)	2m 17s
link_check	2m 8s
build (windows-latest, qt5, 3.14)	4m 47s
build (ubuntu-latest, qt5, 3.10)	2m 33s
build (windows-latest, qt5, pypy-3.11)	11m 21s
build (macos-latest, qt5, pypy-3.11)	5m 5s
build (ubuntu-latest, qt5, 3.14)	2m 32s
build (ubuntu-latest, qt6, 3.12)	2m 40s
build (windows-latest, qt6, 3.11)	4m 10s
build (macos-latest, qt6, 3.12)	3m 11s
build (macos-latest, qt6, 3.10)	3m 24s
build (ubuntu-latest, qt6, 3.10)	2m 42s
build (windows-latest, qt6, pypy-3.11)	7m 46s
build (ubuntu-latest, qt5, pypy-3.11)	3m 34s
build (macos-latest, qt6, 3.11)	2m 48s
build (windows-latest, qt6, 3.13)	4m 19s
build (macos-latest, qt6, pypy-3.11)	5m 31s
build (macos-latest, qt5, 3.10)	2m 58s
build (ubuntu-latest, qt6, 3.11)	2m 33s
build (windows-latest, qt5, 3.10)	4m 33s
build (windows-latest, qt6, 3.10)	6m 23s
build (ubuntu-latest, qt6, pypy-3.11)	3m 34s
build (macos-latest, qt5, 3.14)	2m 47s
build (macos-latest, qt6, 3.13)	3m 10s
build (windows-latest, qt6, 3.12)	4m 11s
build (ubuntu-latest, qt6, 3.13)	2m 37s
Install from SDist and Test	2m 33s
coverage	17s
	1h 55m 27s

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