refactor: remove request_ctx ContextVar, thread Context explicitly #1097

Job	Run time
checks / readme-snippets	10s
checks / pre-commit	37s
checks / test (3.10, locked, windows-latest)	2m 42s
checks / test (3.10, locked, ubuntu-latest)	50s
checks / test (3.13, locked, windows-latest)	2m 20s
checks / test (3.10, lowest-direct, windows-latest)	2m 27s
checks / test (3.12, lowest-direct, windows-latest)	3m 29s
checks / test (3.11, locked, ubuntu-latest)	47s
checks / test (3.11, locked, windows-latest)	2m 32s
checks / test (3.12, locked, ubuntu-latest)	48s
checks / test (3.11, lowest-direct, windows-latest)	2m 51s
checks / test (3.13, lowest-direct, ubuntu-latest)	49s
checks / test (3.11, lowest-direct, ubuntu-latest)	49s
checks / test (3.13, locked, ubuntu-latest)	45s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 11s
checks / test (3.14, lowest-direct, windows-latest)	2m 10s
checks / test (3.14, locked, ubuntu-latest)	2m 12s
checks / test (3.12, lowest-direct, ubuntu-latest)	55s
checks / test (3.12, locked, windows-latest)	3m 29s
checks / test (3.14, locked, windows-latest)	3m 16s
checks / test (3.13, lowest-direct, windows-latest)	2m 36s
checks / test (3.10, lowest-direct, ubuntu-latest)	47s
all-green	3s
	38m 35s

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