refactor: remove request_ctx ContextVar, thread Context explicitly #1096

Job	Run time
checks / pre-commit	29s
checks / readme-snippets	13s
checks / test (3.12, lowest-direct, windows-latest)	4m 26s
checks / test (3.10, locked, windows-latest)	2m 13s
checks / test (3.11, locked, windows-latest)	2m 25s
checks / test (3.10, lowest-direct, windows-latest)	2m 26s
checks / test (3.10, locked, ubuntu-latest)	47s
checks / test (3.12, lowest-direct, ubuntu-latest)	51s
checks / test (3.11, lowest-direct, ubuntu-latest)	49s
checks / test (3.13, locked, windows-latest)	2m 10s
checks / test (3.12, locked, windows-latest)	3m 17s
checks / test (3.13, locked, ubuntu-latest)	47s
checks / test (3.11, lowest-direct, windows-latest)	2m 28s
checks / test (3.12, locked, ubuntu-latest)	51s
checks / test (3.10, lowest-direct, ubuntu-latest)	51s
checks / test (3.13, lowest-direct, windows-latest)	2m 36s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 17s
checks / test (3.11, locked, ubuntu-latest)	51s
checks / test (3.14, locked, windows-latest)	3m 9s
checks / test (3.13, lowest-direct, ubuntu-latest)	43s
checks / test (3.14, locked, ubuntu-latest)	2m 18s
checks / test (3.14, lowest-direct, windows-latest)	3m 19s
all-green	4s
	39m 20s

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