a 1/8" pilot/pin punch is ideal ... it has a nub on the end that sits inside the roll pin to keep the punch centered.

there's also a hollow punch the pin fits into to get it started if it's all the way out, but you don't really need one.

if the pin is in tight, it helps to slide a shim/washer between the link and one of the sides of the plunger when pounding in the pin. That prevents the plunger from sides from squishing together and binding the link. The link should swivel freely on the pin before you put the spring back on.

this is one place: https://www.flippers.com/coil-resistance.html

the flipper coins should be A5141's

the flippers aren't as powerful as a modern game, but they should be able to get the ball to the top of the playfield easily.

typical places to lose energy are:
- crud in the coil sleeve
- worn coil sleeve and/or mushroomed plunger
- crud in the flipper shaft
- end-of-stroke switch misadjusted and opening too early

on big indian, there's two hold relays.

hold relay (R) powers when you first reset the game and stays powered forever unless a lightbox tilt switch opens or the anti-cheat switch opens.

tilt hold relay (H) also powers at reset, but it will unpower if you tilt the machine.

on big indian, the H relay is the one that will cut off the lighting.

both relays should handle being on indefinitely with only the paper wrapper eventually charring and after many years the coil will fail as the insulating varnish breaks down.

both relays will be hot tho ... are you sure that's the source of the burning smell?