What to do about damaged bike frame
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.