Three points:

1. I would replace the defrost timer if I were you. If your fridge isn't working CONSISTANTLY well now that your power has been back on for a few days, then it very well could be that your defrost timer is sticking.

Normally, power is supplied to the fridge's thermostat (aka: "cold control") and if the thermostat wants the fridge to get colder, it's passes that power on to the fridge compressor and evaporator fan. The defrost timer's job is to DIVERT the power from the fridge's thermostat to the defrost heater for 15 to 30 minutes every 20 hours or so to melt all the accumulated frost off of the evaporator coil. That is, a "frost free" fridge is just a fridge that automatically defrosts itself once every day or so.

It's a tiny motor that operates the defrost timer that switches the fridge from "run" mode into "defrost" mode, and friction can cause that tiny motor to stick in the run mode (where the fridge doesn't defrost itself) or the defrost mode (essentially causing the fridge to stop working completely and go into a self induced coma). So, the symptoms of a sticking defrost timer can go from no cooling to no defrosting depending on where the timer is sticking, to anything in between depending on how badly it gets stuck when it sticks.

At 22 years old, if you've never had to replace the defrost timer on this fridge, my feeling is that it would be a good plan to replace it just as preventative maintenance. I wish I had a nickel for every panicked post I've read where someone comes home after work on a Friday afternoon and realizes their fridge isn't working; at all. If that ever happens to you, it's most often caused by the defrost timer sticking in defrost mode. (The defrost timer will have a shaft on it that can be turned manually with a slot screw driver. Since turning the defrost timer the wrong way will damage it, that shaft will be designed like a tamper-proof screw so that it can only be turned in one direction with a screw driver. Unfortunately, every fridge has it's defrost timer in a different location, so it's a good idea to find out where the defrost timer is on your fridge to prepare yourself in advance.)

2. To help you make your decision as to whether or not to fix your fridge, I would check the frost development on the evaporator coil in your fridge. Inside your freezer compartment will be a removable panel that's held on with screws. Often there will be covers over those screws that have to be popped off to gain access to the screw heads. Remove that access panel and you should see an aluminum evaporator coil with fins on it; like this:



That aluminum tubing that goes back and forth is the evaporator coil. It's where the freon evaporates, absorbing heat as it does.

If both the compressor and freon charge are good, then you should see a uniform layer of frost form over the whole length of the evaporator coil, like this:



If, on the other hand, the frost formation isn't uniform, and there's no frost at the downstream end of the evaporator coil, it could just be that the fridge only started running a few minutes before you took the access panel off, but if it doesn't become uniform in 10 or 15 minutes, then the usual cause is either a weak freon charge (most often cuz of a leaking compressor or evaporator), or because the compressor is worn out. Either one of those problems is going to make repairing the fridge uneconomic cuz the repair will cost more than the fridge is worth. So, check out the evaporator coil on your fridge to see if the frost pattern is uniform over the whole evaporator coil (indicating the fridge still has a lot of life left in it), or there's only frost forming at the upstream end of the evaporator (indicating the fridge is already approaching the end of it's useful life). On fridges that are already past the end of their useful life, you often see a "snowball" form at the upstream end of the evaporator, and the rest of the evaporator is free of frost.

3. I see absolutely no reason why the new fridge icemaker couldn't be supplied by the existing water connection if it's compatible. If it's a copper line, I can't see why they couldn't cut the old compression fitting off the end of the line and solder a fitting compatible with the new fridge on. I would ask them to explain their reasoning. Maybe it's because the guys they send out to install their appliances don't know how to solder.