Fuel is a minor portion of launch cost. SpinLaunch is saving fuel but throwing away rockets; it's going to be hard to compete with someone who throws away more fuel but saves the rockets.
If they do pull it off, the logical next step would be to use SpaceX-style reusable rockets as well, in which case you'd save both fuel and rocket costs.
Making the rocket reusable would be the next step, but that's a nontrivial task itself, since it's effectively the upper stage which is the hardest part of reusability. Maybe it's easier for such a small rocket, I don't know. But even then they're just shaving a little fuel off a very low launch cost, in exchange for a 200 lb max payload that has to be hardened to 10,000 gees. The launch cadence would be an advantage today but Starship is supposed to launch three times per day, if the customers are there for it.
I believe the sweating skin idea has been dropped. See this one hour video for a detailed breakdown of the current state of the design:
When I first heard of it, I was thinking, this must be a bad April Fool’s joke, or an incredibly great idea. Turns out, it was the former.
Their goal is to allow the rocket to ignite at 200k ft. Thats only a bit lower than the altitude SpaceX ignites its second stage, which is not reusable
It seems like this launch system will be more expensive per kg, and have far greater loads on the payload than Falcon 9.
I hope it works out, and they find ways to make it commercially viable, but Falcon 9 really has set the bar for success quite high.