You can spin up the Cursor CLI inside the terminal of your IDE of choice and not be tethered to Claude's models.
But if you take speed/availability/cost into account, there might be "better" offers out there. I did some tests w/ windsurf when they announced their swe1 and swe1-lite models, and the -lite could handle easy tasks pretty well. I also tested 4.1-mini and 4.1-nano. There are tasks that I could see them handle reliably enough to make sense (and they're fast, cheap and don't throttle you).