The purpose of the diodes is that they keep the current flowing in one direction--which is critical to many DC circuits, including this one. I'm not properly/traditionally educated in electricity, but in a nutshell: when the coil "fires" (momentarily activates) the electricity is flowing *in one direction* from the power source, through the coil, to ground. Once the power or path to ground is removed though, some of that current can travel backwards FAST--which in this case is leading back to the TIP122/102 transistors, their predrivers, and chips. The diode kinda acts like a check-valve--it only allows the current to move in one direction.
You saw how much force it can have when you saw the transistor blow for the coin door lockout coil. Note the presence of the diode on that coil. Wired up properly (**red wire on banded side of diode**), it shouldn't blow the transistor again. In some cases, the lack of a diode (or improper wiring to the diode) will blow just the TIP122/102 transistor but it's still capable in some cases of taking out the pre-driver and chips. I don't think I've seen too many cases where the chips got hit, and I wouldn't rush to replace the PIA just yet.
Honestly--I'm not sure why the right bumper *isn't* locking on. Perhaps the transistor on that side failed as well, but didn't create a short that could lock on the bumper? I'd investigate the drivers and predrivers on the driver board for both of those bumpers again.
It's possible the shop you bought the coil from doesn't include diodes on their coils, or you simply ordered a coil that doesn't include one installed. Some pinball manufacturers place the diodes elsewhere in the game, so the coils can be wired up either way--not the case with these Williams games though. Double check the resistance on both coils, add a diode to each, make sure red wire is on banded side of the coils.
You're probably gonna need to replace the transistors on the driver board for the pop bumpers again too. I'd definitely check them before turning the game on again. If they're out of spec compared to the working transistors nearby, I'd go ahead and replace them. I know what a pain in the ace it can be having to take the boards out and reinstall them over and over, so make sure to check everything carefully before giving it another shot. At this point I think you probably (hopefully!) have everything you need to get the game working--just need to take the last few steps to make it past the finish line.
Good luck!