client: store server_info on ClientSession after initialization #1460

Job	Run time
checks / readme-snippets	14s
checks / pre-commit	34s
checks / test (3.10, locked, ubuntu-latest)	51s
checks / test (3.11, lowest-direct, ubuntu-latest)	54s
checks / test (3.11, locked, ubuntu-latest)	53s
checks / test (3.10, locked, windows-latest)	2m 27s
checks / test (3.12, lowest-direct, ubuntu-latest)	58s
checks / test (3.11, lowest-direct, windows-latest)	2m 38s
checks / test (3.11, locked, windows-latest)	2m 52s
checks / test (3.10, lowest-direct, windows-latest)	2m 33s
checks / test (3.12, locked, ubuntu-latest)	59s
checks / test (3.12, locked, windows-latest)	3m 27s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 16s
checks / test (3.12, lowest-direct, windows-latest)	3m 55s
checks / test (3.13, lowest-direct, windows-latest)	2m 51s
checks / test (3.14, locked, windows-latest)	3m 30s
checks / test (3.13, lowest-direct, ubuntu-latest)	53s
checks / test (3.14, locked, ubuntu-latest)	2m 26s
checks / test (3.14, lowest-direct, windows-latest)	2m 34s
checks / test (3.13, locked, windows-latest)	2m 23s
checks / test (3.13, locked, ubuntu-latest)	47s
checks / test (3.10, lowest-direct, ubuntu-latest)	1m 1s
all-green	6s
	41m 2s

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