those old G series mowers were notorious for electrical problems. Maybe not so well publicized but the rest of the tractor is great, it's almost always electrical stuff that goes south on them
these require 12v to run (electric lift pump). They'll theoretically run without it but you'd have to feed the injection pump somehow..
starter "helper" relays are a godsend. Basically it's a 30a #87 relay installed in the starter solenoid circuit. Unplug the solenoid trigger wire (white?), and use that to trigger the relay on pin 86. Pin 87 goes to the starter. Pin 30 to the battery (or better/easier, on the starter's big wire that comes from the battery), and pin 85 goes to a good chassis ground. John Deere used to sell a kit specifically for mowers that had similar issues, part # was AM107421 but probably a new number now. It has the terminals already installed so all you do it plug it in and put the terminals where they belong. Super easy. BTW it only works with the "small" starters, some of the earlier G series used the "big" starter. The big starter had an external solenoid and had one large wire on the starter where the late starters were a gear reduction style starter with two wires (one large one small). If you have to replace a starter use the later one and wire it accordingly. That starter uses a lot less power to crank the engine and lasts nearly forever. Mine is still original-from 1994 IIRC.
run through the harness connectors and do a voltage drop test across all of them. Fix the bad ones. There is one below the steering wheel, underneath the tractor, that is almost always in bad shape. Hard to get at too but once you have the harness retention tab bent out of the way, you can get better access. The pins are often corroded. Similar for the voltage regulator connector under the steering wheel, behind the radiator. Almost always corroded if original. I've had to replace a bunch of those. On mine it was so bad that the wires were burnt, so I just removed the dynamo and put a regular 40A alternator on it that came off of a core BX24 engine from work (I worked at a dealer at the time). Not terribly hard to wire that in, and it bolts on-but you need to make sure to get the spacer (if you have seen a BX24 alternator setup, you know). There is a spacer on the pivot bolt and it is required. That or a stack of washers, it aligns the belt properly. It charges a little better at low speed (near idle) but it also needs a few wiring changes to be made at the reguator/rectifier in order to be done properly, and to run a good 10ga wire from the lug on the alterator to the starter, with a 50A slow-blow fuse inline. The only advantage is that it will charge a little better at lower engine speeds. The factory installed dynamo is plenty of charging power for the application though, and simpler. Not much goes wrong with them, maybe a bearing once in a while and they are simple to rebuild.