Major pitting in bearing cups; what can I replace?

The kind of side-ways wobble is quite normal for freewheels of the period and should not cause trouble. But it's also a sign that the axle is not perfectly straight which also overloads the bearings of the hub. Enough to cause wear.

It may be possible to replace the cones and the ball-bearings but not the races which were rarely replaced. If you also replace the axle, this could solve your problem. Your LBS if it has been there for long could still have cones & bearings. It will probably as expensive in parts and labour as a new cheap rear wheel with a free-hub and fit a 7-speed cassette as they still exist. Free-hub allow for a better load distribution on the rear axle as the bearings are further apart.

Just check whether the rear triangle is for a 130mm spaced hub not 126mm that was also common around the 1990s. Your LBS may easily correct the spacing on a steel frame.

You should replace the balls for sure. They're cheap, readily available (from any bike shop -- just take in a sample ball), and would very likely solve most of your problems.

You can also fairly easily replace the cones. You might have to order them, but they should be reasonably inexpensive. You may want to replace the axles at the same time (they often come as a set).

Replacing the cups would require replacing the hubs, and, if you hire it done, this is usually more expensive than just replacing the wheels. But you can likely get several years of service out of the bike with the existing hubs.

Extracting the cup is possible, but tedious.

After checking the bearing races, I saw several defects on race lines. Since that means either the wheel or the cup must be replaced, I had nothing to lose by trying to extract it. In this hub, it's impossible to push out the cup with a steel rod from the other side, because the inner diameter is too small. I have cut a slot in the bearing cup with a rotary tool, making it possible to compress it a little bit like a c-clip, and several horizontal slots that where I could put a screwdriver and use it as a leverage. Eventually, the cup was out. I damaged the aluminium body a bit, which is not good. So for everyone attempting this - watch for the line separating steel and aluminium, it's barely visible but it's there, the bottom surface isn't smooth.

This article has some information on replacing bearing cups:

http://bruceteakle.blogspot.com/2017/03/replacing-bicycle-wheel-bearing-hub-cups.html

I will now try to find either a cup or a used hub that uses same type of cups.

Thank you everyone for advice!