While looking for a schematic I ran across this old thread in which several people mention the same problem with different pong systems. (It's odd that it's always one of the paddles and never a ball or the playfield)
https://forum.digitpress.com/forum/s...gely-need-help!

No verified solution though, but it does show a general pong schematic.

All of the schematics I've found for pong systems show diodes instead of NAND gates, including that thread, though it does mention that some systems use NAND gates instead of diodes. The idea is the same though, take several different signals, one for each "object" to be displayed, and mix them with the sync signal to get your video signal. So the problem has to be somewhere between that paddles output and the point where they are mixed. Since you jumped the signal wit no effect then it's probably either the signal itself(not as likely if you tried a different chip and had the same problem) or in the output of the NAND gate.

A logic probe is just a device that shows if a signal is high or low or pulsing(which is probably what you'd be looking for). Or a signal tracer would work better. Some higher end meters include a signal tracer setting. Or an oscilloscope would be even better. If you compare the path of the good paddle to the bad paddle you should be able to see where it drops out.