There's some precedent for chopping out the GC for short-lived applications. DMD, the (self-hosting) D compiler, does this, effectively using a push-only stack for the heap, never doing anything akin to freeing memory. [0]
In modern GCs, allocation is already as fast as can be (pointer-bump allocation), so I imagine the only win in chopping out the GC is that you don't need to initialize the GC (it's otherwise roughly equivalent to simply terminating before the GC needs to be invoked).
Perhaps the DMD example isn't quite the same, though, as it's possible its GC has slower allocation than pointer-bump.
I would look into GraalVM native images if you want fast startup.