Status Quo are renowned for using as few chords as possible - they called one of their albums In search of the fourth chord!

That "Wind and Rain" is in C (Johnny C.'s version was in D (and sung an octave or two lower)).

Answering my own question (with a little help from my friends!), I think I've been working on the assumption that chords with accidentals in would sound "wrong". Actually they just sound "different" - just as a scale in different modes sounds different. Presumably there are some combinations of notes which sound so "different" that you'd never actually want to use them - just as (I seem to remember) there are a couple of modes which are more or less theoretical & have no real-world use - but I guess you'd have to bend a 1-3-5 triad quite a long way out of shape before you got an actual dischord.