No need to be nervous, these kinds of hacks are common in vintage EM games.
If you want, you can remove the hack jumper and see what coils stop working on the PF. Then you can look for the break in the red daisy-chain wire, and fix it. The break is most likely at one of the coil solder lugs, which could be at any coil on the PF that uses the red wire for the power supply.
If all the coils continue to work, then you should just leave that hack jumper off the machine. On the other hand, if some coils do go dead, and you can't find the break in the red daisy-chain, you can always put a jumper wire back on your game, at the same place. Use a thicker gauge wire, and do a better job of soldering, and you'll be that much better off.
Note that if you focus on this problem, you'll be spending your time on a totally different problem than fixing your stuck 300/3-ADV relay issue. That's not a bad thing, it's just a personal decision. You may want to keep working on your stuck relay issue for now, and deal with the hack jumper later, when your current issue is resolved.
As for the alternator relay not working, if the coil is out of the circuit, you will not get an ohm reading across a coil if the coil is bad (open). A bad coil will either be open (infinite resistance) or dead shorted (1 ohm or less). HOWEVER, you can ONLY check the ohm reading across a coil by disconnecting one of the coil leads, and checking the coil out of circuit. If you check a coil in circuit, you are no longer assured that you are only measuring the resistance of the coil, and you can be fooled by seeing a reasonable resistance reading, even on a totally open coil.
If the alternator coil is good, but it now won't energize when the 00-90 switch is closed, try wiggling the jones plugs in the sockets, and maybe slightly pulling up and reseating the plugs, for the cables you removed when you unplugged the PF. Bally jones plugs of this era are notorious for having connection problems. Also make sure you got all the cables plugged solidly back in. Checking (and re-seating) the jones plugs for the cables plugged into the head wouldn't hurt either.