refactor: remove request_ctx ContextVar, thread Context explicitly #1099

Job	Run time
checks / readme-snippets	13s
checks / pre-commit	38s
checks / test (3.10, locked, ubuntu-latest)	49s
checks / test (3.10, lowest-direct, ubuntu-latest)	56s
checks / test (3.11, lowest-direct, windows-latest)	2m 41s
checks / test (3.12, locked, ubuntu-latest)	53s
checks / test (3.11, lowest-direct, ubuntu-latest)	57s
checks / test (3.10, lowest-direct, windows-latest)	2m 38s
checks / test (3.11, locked, windows-latest)	2m 13s
checks / test (3.10, locked, windows-latest)	2m 22s
checks / test (3.11, locked, ubuntu-latest)	48s
checks / test (3.12, locked, windows-latest)	3m 40s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 16s
checks / test (3.12, lowest-direct, windows-latest)	4m 2s
checks / test (3.14, lowest-direct, windows-latest)	2m 40s
checks / test (3.12, lowest-direct, ubuntu-latest)	56s
checks / test (3.13, lowest-direct, windows-latest)	2m 37s
checks / test (3.13, locked, windows-latest)	2m 35s
checks / test (3.13, locked, ubuntu-latest)	46s
checks / test (3.13, lowest-direct, ubuntu-latest)	49s
checks / test (3.14, locked, windows-latest)	4m 59s
checks / test (3.14, locked, ubuntu-latest)	2m 28s
all-green	3s
	41m 59s

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