Their next-gen ones should be reusable, and share a lot of design with the Falcon 9 family. Methalox might next, as it's very promising, but the RP1 supply chain is well established.
It's not just that the Shuttle was expensive to refurbish. It was also very expensive to build.
Whereas Falcon 9 is much less expensive to build new than Ariane 6.
Ariane 6 was designed to be cheaper to build (cheaper than Ariane 5, that is).
For context, my understanding is that SpaceX builds one new Falcon 9 upper stage (not reused) every day. I doubt there is another entity on earth building a new rocket every month.
SpaceX has a large fleet of Falcon 9 reusable boosters, but still needs to build one every now and then.
Probably not quite that often. They're aiming to launch ~150 this year (although with the recent RUD, that'll be pretty tough). But yeah - 1 every other day is still very impressive.