I recently bought a used bike from a police auction that had a nasty cut in the frame (presumably caused by a angle grinder or something similar). Should I be worried about the bike's structural rigidity? If so, what should I do to mend the frame? I thought about wrapping a piece of aluminum around the damaged area and bolting it down with metal ties, but I'm not sure how much that'd help.
I personally would not ride that bike. The top tube is severely compromised now, it will get worse as cracks develop from the ends of the cuts.
No-one plans to crash so you can't resolve to always ride it 'gently', You do not want to ride over a pothole or off a curb and have the top tube buckle up into your face, neck or chest.
Your proposed fix of clamping a sheet of aluminium over the tube would not be sufficient, neither will JB weld. There is no safe solution for $30.
Unfortunately, a proper welding job will probably cost more than you paid for the bike.