If you haven't figured it out already, be very careful with traces on Asteroids (and many other Atari boards of the same era). It is VERY EASY to lift traces on these.
As mentioned, using flush cutters to cut the legs, then use needle nose pliers and a soldering iron to heat up and remove the cut leg, then apply some fresh solder and desolder the hole clear. That will give you the best chance of not lifting a trace. I've repaired a couple of Asteroids boards with lifted traces. It is tedious work to fix them from lifted traces, so best not to have any lift in the first place.
A logic probe is a great tool for looking for lines stuck high or low.... especially if you have a good board to compare against as well.
In lieu of that, if you have a DVM with a diode setting, you can with the board powered off and removed, check for shorts to GND or +5V on any suspect chip. Most chips will read a lower voltage drop in one direction, and a higher one in the other direction (like a diode), but what you are looking for are legs that are wide open or shorted. I don't use this technique anymore, but the first PCB I fixed (Space Invaders), I did this and found 2 bad chips, replaced them and it fixed the board. I probed the entire board, so it was tedious, and not an educated means of finding a fault, but it worked.
If you do find a short to ground or +5V on a chip, be aware sometimes legs are grounded or pulled high.... you do need to check the schematic before you make a decision that this might be an actual fault. Also, as chip outputs are tied to chip inputs, if you find a short on an input of one chip, it is more likely it is a short on the output of a chip feeding it. Though inputs can short as well. Worst case you end up replacing 2 chips when only one was bad.
Good luck finding your problem.