All the old carbureted 2-strokes I’ve used tended to run very rich at low rpms, some to the point where if you didn’t rev them often, the plugs would foul. Was this common/necessary on some/all carbureted 2-strokes, and why?

Plug fouling was a common problem with two strokes. They always needed to be changed more often.

Old racing two stroke engines fouled very easily at low revs, as they were set rich to enable them to start and run at low revs, below the designed power band at high revs where they ran efficiently.

It depends on the design, reed valve engines tended to have less problems than piston ported or rotary valve engines.

Modern synthetic oils at a higher mix ratio often helps to reduce soot and fouling. Please note you can’t mix vegetable oil (Castrol R etc) with synthetic oil.

The other problem is worn or badly setup carburettors, particularly worn slide needles and jets which produce a richer mixture.

FI/DI was tried in the past without mainstream success, and development largely stopped due to the difficulty of meeting emission regulations.

However, KTM make fuel injected off road two stroke bikes, so it can be done.