refactor: remove request_ctx ContextVar, thread Context explicitly #1095

Job	Run time
checks / readme-snippets	16s
checks / pre-commit	33s
checks / test (3.11, locked, windows-latest)	2m 28s
checks / test (3.10, locked, ubuntu-latest)	50s
checks / test (3.11, lowest-direct, ubuntu-latest)	49s
checks / test (3.11, locked, ubuntu-latest)	47s
checks / test (3.10, lowest-direct, windows-latest)	2m 45s
checks / test (3.12, lowest-direct, ubuntu-latest)	54s
checks / test (3.10, lowest-direct, ubuntu-latest)	43s
checks / test (3.11, lowest-direct, windows-latest)	2m 59s
checks / test (3.12, locked, ubuntu-latest)	51s
checks / test (3.12, locked, windows-latest)	3m 21s
checks / test (3.12, lowest-direct, windows-latest)	3m 39s
checks / test (3.14, locked, ubuntu-latest)	2m 21s
checks / test (3.10, locked, windows-latest)	2m 20s
checks / test (3.13, locked, windows-latest)	2m 15s
checks / test (3.13, locked, ubuntu-latest)	46s
checks / test (3.14, locked, windows-latest)	3m 11s
checks / test (3.13, lowest-direct, windows-latest)	2m 34s
checks / test (3.13, lowest-direct, ubuntu-latest)	51s
checks / test (3.14, lowest-direct, windows-latest)	2m 27s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 10s
all-green	4s
	38m 54s

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