I disagree with buying a new wheel. If you do that you would still need to buy a freewheel separately, so that doesn't solve the problem.
4 months is enough time to wear out a couple of freewheel cogs if the bike is used a lot, the drivetrain is never lubed, the parts are not good quality, and the rider only uses those couple of gears. I don't know if that's what happened here but it looks like it.
Many (but sure, not all) bike shops will be able to service a bike that is 30 years old, particularly if it was decent quality and has parts that were widely used at the time. The bigger question is whether it will be worth it. Just like moto shops there is a broad spectrum of shops and quality of service. And just like moto shops, good bike shops are becoming rare and deserve your support if you find one.
With bikes, you truly get what you pay for. There are diminishing returns after a certain point, but that doesn't happen until you're into several thousands of dollars.
I was a shop wrench/manager for 15 years and am still involved in the industry, and I'm not the only knowledgeable person here telling you what's up. The last couple of cogs on that freewheel are shot. If the freewheel was still together you might get lucky and find a shop with a couple of replacement cogs, but that's pretty rare these days. Regardless, since it fell apart it's not worth the time or effort trying to rebuild it. Take it back to where you got it or suck it up and buy some new parts. Sorry/not sorry.