Re-integrate Sphinx's linkcheck into CI configuration #5924

Job	Run time
package	15s
build (windows-py39-unittestextras)	23m 14s
build (windows-py39-pluggy)	7m 21s
build (windows-py39-xdist)	6m 33s
build (windows-py310)	6m 51s
build (windows-py311)	14m 46s
build (windows-py312)	16m 10s
build (windows-py313)	16m 46s
build (ubuntu-py39-lsof-numpy-pexpect)	27m 16s
build (ubuntu-py39-pluggy)	5m 14s
build (ubuntu-py39-freeze)	44s
build (ubuntu-py39-xdist)	5m 30s
build (ubuntu-py310-xdist)	5m 22s
build (ubuntu-py311)	15m 20s
build (ubuntu-py312)	18m 48s
build (ubuntu-py313-pexpect)	24m 49s
build (ubuntu-pypy3-xdist)	7m 40s
build (macos-py39)	4m 54s
build (macos-py310)	4m 30s
build (macos-py312)	5m 14s
build (macos-py313)	4m 3s
build (doctesting)	47s
build (plugins)	57s
check	2s
	3h 43m 6s

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