I think there is a basic confusion here (which may also pertain to the agent OP dealt with).

The "gate" in "gate check" refers to where the bag is checked/accepted (in contrast to the usual check-in counter), not where it is delivered. It is still treated as a normal checked bag (with a barcoded tag) and thus delivered to baggage claim at the final destination.

The other term is "valet check", which is offered as a substitute service for bags that would normally be carry-on, in the case of regional planes (Eagle) whose bins simply cannot accommodate such bags. These receive simple, nondescript valet tags and are situated in the plane's baggage hold for easy access immediately after landing so they can be delivered to passengers planeside.

Mainline planes (e.g., those operating PHX-SFO) do not offer valet check, because most people can bring their carry-ons with them; the need to gate check is relatively infrequent. This is in contrast to small Eagle planes, where a valet service is necessary for everyone who has a full-size carry-on bag that they wish to keep accessible as much as possible during their trip.