A good Moon base should for sure be bigger than an orbital outpost; unless we have an awful lot of traffic there, on the orbit around the Moon... but we're not there yet.
However, to completely avoid having a Moon orbiting facility would be to err to the other direction. Suppose, for example, you have an emergency on the Moon. If you need to fly to Earth, it's easier to meet an Earth-bound ship in the Moon orbit than to have that ship landed on Moon; reminding, SpaceX lunar ship isn't intended to fly to Earth, with good reasons. Or if you need an urgent delivery of something unique from the Earth, it's easier to deliver to the Moon orbiting outpost, rather than to have it with Moon landing capabilities. If you need delivery of materials from the Moon, a convenient place is to accumulate them in orbit; Earth's history of sea shipments suggests that. Of course there could be other reasons.
Size of Starship doesn't really matter - Gateway could be smaller, and still quite useful.
Gateway will be almost completely useless for this. Gateway is in a very odd orbit, a near rectilinear halo orbit. that NRHO spends most of its time a massive distance from the moon. Most of the time the Earth is closer to the moon than the gateway is.
You can make a good argument for having a station in low lunar orbit, but NRHO is really silly. It was only chosen because it's all the Senate boondoggle SLS could reach.
Edit: NRHO is only close to the moon once every 7 days, versus a low lunar orbit which orbits the moon in 2 hours.
Maybe they could use a Starship as an additional module for Gateway or the ISS? Just dock it to one of the ports and leave it there permanently. Maybe even send up a special one with some extra docking ports on it. You could even convert the fuel tanks into extra cabin space (the classic "wet workshop" space station design).
Gateway should have at least 125m³ from what I saw on its Wikipedia page.