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You don't just need one starship to do one moon flyby. You need a bunch of starships, and you need in-orbit refueling to work well, and you need different starship designs (a "fuel tanker" upper stage, versus the normal upper stage that you are sending to the moon). You probably also need a bunch of custom software and hardware for visiting the moon, that you don't need for low-earth orbit (doing TLI propulsion burns, antennae for long-distance communication, etc). Going to the moon is going to be way, way down on the priority list of things to test, even once Starship is working in a minimum-viable-product capacity.

@JacksonWagner not 100% sure, but I believe Starship can do a lunar flyby without refueling if the payload it's carrying is low enough. Dear Moon shouldn't have much payload at all, and if they launch/land from earth in a dragon capsule then the Starship doesn't need flaps, a heatshield, or header tanks. That gets 250 tons to LEO, or a lot of spare delta V if you have a small payload.

I still don't think it will fly in 2023, but mainly because of the extra hardware needed (docking port, windows, life support) than because it needs to refuel.

More precise figures: if you use an expendable Starship (no heat shield, flaps, or header tanks) then it can send about 60 tons to TLI in a single launch. A reusable Starship can't reach TLI in a single launch.

https://twitter.com/Neo_EimajOzear/status/1621248854350864384