That's why you basically always need a rocket engine anyway.
It's physically possible but the engineering is insanely difficult, much more so than a traditional rocket, which is already no cakewalk.
I'm not saying it's impossible, just that to pull this off while still having a craft capable of reaching orbital speeds is an immense challenge.
Also, economic viability: per the Tsiolkovsky rocket equation, for a 100kg payload (almost microsat territory) and an orbital-insertion engine with a generous ISP of 320, you'd still need over 500kg of propellant to accelerate the "second stage" 5km/s to orbital velocity. This ignores the mass of the tank and engine and electronics (which have to survive 10,000g!). If you remove the latter from the payload you're looking at just a few kg remaining for the sat itself and you've massively raised the cost for the upper stage due to the exotic materials required to hold up under such acceleration. Then even if SpinLaunch or another team pulls off said engineering, you have to compete against a traditional rocket which could launch ~100 equivalent satellites to low-Earth orbit in one shot—at a known cost, with high reliability—using only materials and design widely in use today.
Edit: The same numbers with a 1,000kg payload are ~6,000kg at launch. How are you going to design a 6,000kg craft which can survive 10,000g?
Merlin 1D and Rutherford get close to 350 in vacuum with RP-1. Hydralox would need to be insulated during spin up, which probably makes it prohibitive. In any case, if anything goes wrong in spin-up or deployment, the entire facility is toast. Whether there's a bit of extra chemical energy added to the mix is likely a rounding error.
Again, this is because the vehicle can be accelerated electrically under vacuum until it reaches the required velocity. The heavy mass (in the vehicle’s nose cone) is not part of the orbital vehicle.
The nose is copper and aluminum that do melt under these temperatures, but this is prevented by two elements.
Only the first five seconds of the launch will deliver the majority of heat, and this heat can be dissipated by thermal conductivity.
I think they have this covered.
> [...] the front of the craft would be white-hot [...]
To the extent heating is inevitably experienced, a conventional way is to get rid of it by radiative cooling. Complaints about the visual appearance of a radiative cooling subsystem visibly doing its job is a bit weird.