refactor: remove request_ctx ContextVar, thread Context explicitly #1094

Job	Run time
checks / pre-commit	34s
checks / readme-snippets	11s
checks / test (3.10, locked, ubuntu-latest)	51s
checks / test (3.12, locked, ubuntu-latest)	57s
checks / test (3.10, lowest-direct, windows-latest)	2m 14s
checks / test (3.12, lowest-direct, ubuntu-latest)	59s
checks / test (3.12, lowest-direct, windows-latest)	3m 43s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 16s
checks / test (3.13, locked, windows-latest)	2m 18s
checks / test (3.11, lowest-direct, ubuntu-latest)	54s
checks / test (3.13, lowest-direct, ubuntu-latest)	50s
checks / test (3.10, lowest-direct, ubuntu-latest)	56s
checks / test (3.12, locked, windows-latest)	6m 38s
checks / test (3.13, lowest-direct, windows-latest)	2m 34s
checks / test (3.10, locked, windows-latest)	2m 14s
checks / test (3.11, locked, ubuntu-latest)	53s
checks / test (3.11, lowest-direct, windows-latest)	2m 53s
checks / test (3.11, locked, windows-latest)	2m 24s
checks / test (3.13, locked, ubuntu-latest)	51s
checks / test (3.14, lowest-direct, windows-latest)	2m 16s
checks / test (3.14, locked, windows-latest)	3m 26s
checks / test (3.14, locked, ubuntu-latest)	2m 19s
all-green	3s
	42m 14s

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