He’s partially correct. See the definition of “chief mate” in 46 CFR 10.107:
Chief mate means the deck officer next in rank to the master and upon whom the command of the vessel will fall in the event of incapacity of the master.
But, being the Master doesn’t meet that definition for chief mate, so the Master cannot be a “chief mate” and strict application of would mean it doesn’t meet the requirements for service as Chief Mate. And in your example, if you are serving as AB but hold a license, you are not a “deck officer” on that vessel, and also could not be the “chief mate” even if there are no mates on the vessel.
But you are right as to the endorsement he asked about, Master Unlimited Inland. There is no provision to qualify for this endorsement using chief mate service in 46 CFR 11.435. So the regulatory definition of a chief mate doesn’t apply. It is also likely that the requirement for inland or Great Lakes service will be strictly applied, i.e. time on near coastal or oceans won’t count.
This is correct for Master Unlimited Oceans or near coastal. You only “avoid” the exam if you are upgrading from a license as Chief Mate Unlimited. Not serving as a “chief mate” , but holding the license. See 46 CFR 11.903( c)(1). An exam is also required for Master Unlimited Inland regardless of what you already hold, see 46 CFR 11.903(a)(1).