That's why spinlaunch is flinging a whole rocket high into the air rather than just the payload.

This isn't the case at all. The trap your reasoning falls into is assuming that for some reason an object must be in an orbit after being thrown. Obviously, an orbit comes back to it's original position at some point. In many cases atmospheric drag converts what would be an orbit into burning up in the atmospheric or smashing into the ocean somewhere.

But if you throw something hard enough from the Earth's surface, it absolutely does not have to return to that position. You would just need to throw it hard enough that it was at escape velocity. Due to air friction, the actual speed you would need to throw the object would be flat out absurd if on Earth at sea level. But on a body like the moon with no atmosphere it isn't that bad at all. The bonus to this is the direction doesn't matter at all really. Anything other than straight down is fine.

Now, where this does become problematic is when the velocity you would need to achieve is higher than the speed of light. At that point you're basically on a neutron star or some supermassive planet.

> Now, where this does become problematic is when the velocity you would need to achieve is higher than the speed of light.

Isn’t this sort of a definition of a black hole? My non-physicist intuition tells me a neutron star can’t have an escape velocity higher than the speed of light, otherwise the light wouldn’t escape

No one here is talking about Spinlaunch achieving Earth escape velocity, only orbital velocity.

We're not talking about escape velocity though. (And if we're going to nitpick, even on an escape trajectory, the launch point is still on the hyperbola.)