Tim Streater formulated the question :
I'd say cars are too complex to do much of that any more.

They are more complex, but the complexity in many ways helps you
diagnose the problems.

I used to
tinker with my Mini 50 years ago and mostly made a mess of it, so I
gave up.

They never came any simpler, than a mini.

Can you even buy workshop manuals for today's cars?

You can, but the home mechanic manuals skim most of the detail. Full
manufacturers manuals are much more detailed.

Don't you
need a computer to check everything? The garage will have one but
would the right kit be available to an end-user?

For not much money, you can buy diagnostic units - complete stand alone
units, or interfaces and software for a laptop. I have four. The car
basically is able to tell you, or at least provide some clues as to
what ails it. How much simpler can it be made?