feat: store server_info on ClientSession during initialization #1130

Job	Run time
checks / pre-commit	31s
checks / readme-snippets	15s
checks / test (3.11, lowest-direct, windows-latest)	2m 35s
checks / test (3.10, lowest-direct, windows-latest)	2m 44s
checks / test (3.11, locked, windows-latest)	2m 30s
checks / test (3.10, locked, ubuntu-latest)	53s
checks / test (3.11, lowest-direct, ubuntu-latest)	59s
checks / test (3.12, locked, windows-latest)	3m 48s
checks / test (3.12, lowest-direct, windows-latest)	5m 42s
checks / test (3.10, locked, windows-latest)	2m 11s
checks / test (3.13, lowest-direct, ubuntu-latest)	56s
checks / test (3.10, lowest-direct, ubuntu-latest)	57s
checks / test (3.11, locked, ubuntu-latest)	51s
checks / test (3.12, locked, ubuntu-latest)	1m 2s
checks / test (3.12, lowest-direct, ubuntu-latest)	1m 5s
checks / test (3.14, locked, windows-latest)	3m 16s
checks / test (3.13, locked, windows-latest)	2m 37s
checks / test (3.13, lowest-direct, windows-latest)	2m 20s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 20s
checks / test (3.13, locked, ubuntu-latest)	49s
checks / test (3.14, locked, ubuntu-latest)	2m 23s
checks / test (3.14, lowest-direct, windows-latest)	2m 22s
all-green	4s
	42m 10s

Provide feedback