My experience in this is that painters paint. At best the painter may
have to repair dings and the like from other trades doing their thing.

Runinng tape and mudding is the job of the drywall contractor. Maybe he
subs out one of the two, or has separate crews for each, but it's part
of his job.

No builder I've ever dealt with expects the painter to tape and mud.
What your painter is telling you is correct.