How do you check for problems on a car that doesn’t have electronics or a computer? My guess is to do a visual inspection of the car to see if there are any problems.
I guess you know you're old when you see people asking questions like this and wonder why anyone would have to ask them. 🤔
I do this all the time because I still have a few older cars where the only electronics are the alternator and the radio. Having worked on cars for over 50 years (though strictly on an amateur basis) it's something I don't really even have to think about, I just go at it.
Basically, you take your knowledge of what conditions would likely cause what symptoms and start investigating the appropriate items to see where it leads you. There are instruments you can use such as dwell/tach meter, timing light, vacuum gauge, etc., and if you're a pro you'd have things like an ignition analyzer with a scope that can thoroughly check distributor operation.
https://www.youtube.com/watch?v=lEDMuzj4GKM
Well you would have to do things as you would back then when there was no computers on cars. Yes that also includes visually inspecting things and manually adjusting things.
The same way you look after a washer, lawnmower, or any other machine.