I think their basic idea is sound, it'll probably eventually function. My concerns are if it'll be worth it, since there's a trade-off being made between fuel mass and heat shield mass, and at least based on Scott Manley's summary video this week, the rocket with the heat shield is close to the same mass as the Electron rocket. In which case I'm not sure the complexity of SpinLaunch is worth it.
NASA can probably trade the tech, if it even barely works, for budget from the Pentagon. This is a ballistic launch system without the tell of a rocket burn.
I haven't seen Scott Manley's video; but if the mass cost of a heatshield is so significant that's a problem.
However, there's a benefit in this case: heat shield probably scales with area (often something like mass^(2/3)), making it progressively less significant compared to fuel mass (which is roughly a constant fraction, i.e. it scales like mass^(1)). I think a high altitude launch site could make a significant difference as well (although that creates other logistic inconveniences). Atmospheric pressure approximately halves every 5km, and air resistance is roughly proportional to pressure.
Building and SpinLaunching at above 10k feet (3km) seems like a no-brainer. It reduces heating and weight while increasing terminal altitude substantially... like 2x.
It would be more efficient, if they could use a whip effect to (match impedance) capture much more of the rotational energy they put into the spin, more like a trebuchet.
Edit: the video in question: https://www.youtube.com/watch?v=8Phy3n_S3ng
and a more honest and detailed look at the company, again from Scott: https://www.youtube.com/watch?v=JAczd3mt3X0