feat: store server_info on ClientSession during initialization #1185

Job	Run time
checks / readme-snippets	11s
checks / pre-commit	33s
checks / test (3.11, lowest-direct, windows-latest)	2m 41s
checks / test (3.12, lowest-direct, ubuntu-latest)	1m 0s
checks / test (3.10, locked, windows-latest)	2m 31s
checks / test (3.10, locked, ubuntu-latest)	55s
checks / test (3.10, lowest-direct, windows-latest)	2m 20s
checks / test (3.11, locked, windows-latest)	3m 7s
checks / test (3.12, locked, ubuntu-latest)	53s
checks / test (3.12, lowest-direct, windows-latest)	3m 44s
checks / test (3.13, lowest-direct, ubuntu-latest)	56s
checks / test (3.12, locked, windows-latest)	3m 18s
checks / test (3.11, lowest-direct, ubuntu-latest)	56s
checks / test (3.10, lowest-direct, ubuntu-latest)	58s
checks / test (3.13, locked, ubuntu-latest)	48s
checks / test (3.13, locked, windows-latest)	2m 19s
checks / test (3.13, lowest-direct, windows-latest)	2m 48s
checks / test (3.14, locked, windows-latest)	3m 5s
checks / test (3.14, lowest-direct, windows-latest)	2m 57s
checks / test (3.11, locked, ubuntu-latest)	51s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 20s
checks / test (3.14, locked, ubuntu-latest)	2m 28s
all-green	4s
	40m 43s

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