As an intuitive answer to "why can't it result in a smaller circular orbit":
The collision occurs at radius $r$. Therefore the resulting trajectory must pass through a point $r$ distance from the central body.
A hypothetical smaller circular orbit would have some radius $r'$. But a circular orbit by definition has constant radius. A body moving in that orbit is always at $r'$ distance, never at $r$. So for an instantaneous collision to produce a radius $r'$ circular orbit, the body would be required to teleport from distance $r$ to distance $r'$. That's clearly impossible.
So that means that if the body is to end up in a circular orbit at $r'$, there must be some intermediate non-circular trajectory taking the body from radius $r$ to radius $r'$, before it can take up its final circular orbit. But if the impact puts it on such a trajectory, why does it later change trajectory again (from a non-circular orbit into a circular one) when it reaches radius $r'$?
For this to happen would require another force to cause the change in trajectory; another impact, or an engine firing, etc. But that would mean the lower circular orbit at radius $r'$ isn't the result of the initial collision, but the result of both events (taking place at different times and places). If the impact is the only event we're analysing, then the "intermediate" trajectory is the final one.
Basically: an object can never transition from a circular orbit at one radius to a different circular orbit at another radius (whether lower or higher) as the result of a single event. And the reason is quite straightforward: the object is simply not located on the second circle, so it can't possibly be in a circular orbit on that circle!