Improve MultiIndex label rename checks #24230

Job	Run time
Linux-Musl	2m 45s
Linux-32-bit	2m 45s
python-dev	0s
windows-latest actions-310.yaml	2m 48s
macos-14 actions-312.yaml	3m 1s
macos-14 actions-313.yaml	3m 3s
macos-13 actions-312.yaml	3m 6s
windows-latest actions-311.yaml	2m 47s
Pyodide build	3m 0s
macos-13 actions-313.yaml	3m 5s
macos-13 actions-310.yaml	3m 5s
windows-latest actions-313.yaml	2m 49s
macos-14 actions-311.yaml	3m 0s
windows-latest actions-312.yaml	2m 48s
macos-14 actions-310.yaml	3m 6s
macos-13 actions-311.yaml	3m 9s
ubuntu-24.04-arm actions-313.yaml	3m 0s
Pyarrow Nightly	3m 0s
Freethreading	3m 0s
ubuntu-24.04 actions-313.yaml	3m 2s
Minimum Versions	2m 59s
Future infer strings	3m 0s
Locale: it_IT	3m 0s
ubuntu-24.04 actions-312.yaml	3m 1s
ubuntu-24.04 actions-310.yaml	2m 59s
ubuntu-24.04-arm actions-312.yaml	3m 15s
ubuntu-24.04-arm actions-311.yaml	3m 0s
ubuntu-24.04 actions-311.yaml	3m 0s
Downstream Compat	3m 0s
ubuntu-24.04-arm actions-310.yaml	3m 0s
Numpy Dev	3m 0s
Pypy	1m 31s
Future infer strings (without pyarrow)	3m 0s
Locale: zh_CN	3m 0s
	1h 37m 4s

Provide feedback