Is it ok to use caulking instead of tape and mud on my new drywall ceiling?
No no no no and NO.    You need to fire this guy immediately.   Anyone who even brings this up is beyond reproach as far as incompetency in this area.

Caulk is expensive.
Caulk cannot be smoothed to seamlessly fade into drywall.   Even if you took days/months to flatten caulk it would still have seams and ridges.
Caulk will not look the same when painted as drywall.   That is the whole point of using mud/joint compound... when you paint it, it looks the same as drywall.
What is he supposing for any nicks or scratches?   More caulk?
Caulk will dry out over time and harden and loosen.

However... I have started using caulk for ceiling cracks that keep appearing.   It is a good binder for seasonal contraction - but I tape and put joint compound over the caulk!
Edit: I have a small room in my basement to finish in 2-3 months... Thinking about giving the caulk a go.   I don't mind the caulking but to get the pictures right I would need to put on a little primer and paint and those steps are a PITA to do over again...   I will try to do a couple feet of a seam like this.