refactor: remove request_ctx ContextVar, thread Context explicitly #1098

Job	Run time
checks / readme-snippets	12s
checks / pre-commit	32s
checks / test (3.11, lowest-direct, windows-latest)	2m 23s
checks / test (3.10, locked, ubuntu-latest)	50s
checks / test (3.10, lowest-direct, ubuntu-latest)	57s
checks / test (3.10, locked, windows-latest)	2m 4s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 20s
checks / test (3.11, locked, windows-latest)	2m 32s
checks / test (3.13, lowest-direct, ubuntu-latest)	50s
checks / test (3.11, lowest-direct, ubuntu-latest)	51s
checks / test (3.14, lowest-direct, windows-latest)	2m 23s
checks / test (3.11, locked, ubuntu-latest)	46s
checks / test (3.12, locked, ubuntu-latest)	55s
checks / test (3.13, locked, windows-latest)	2m 33s
checks / test (3.13, locked, ubuntu-latest)	46s
checks / test (3.14, locked, ubuntu-latest)	2m 18s
checks / test (3.13, lowest-direct, windows-latest)	2m 35s
checks / test (3.14, locked, windows-latest)	3m 9s
checks / test (3.12, lowest-direct, windows-latest)	3m 41s
checks / test (3.12, lowest-direct, ubuntu-latest)	58s
checks / test (3.12, locked, windows-latest)	3m 37s
checks / test (3.10, lowest-direct, windows-latest)	3m 8s
all-green	5s
	39m 25s

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