we are, in many cases, quite well served by theories which are known to be false.This is a mistake! Consider a theory of motion, say, which we'll call T. We know T is false, but it's also a good approximation to the truth in common and well defined circumstances.
We do not use theory T. We use theory U which consists of what I said in the first paragraph: that theory T is an approximation, and useful in certain circumstances. Theory U contains in it theory T, but also some other ideas including the refutation of T. Theory U is a way of approximating motion in certain circumstances, it's useful, and it's not known to be false. Theory U is just plain better.
If we can't create a true variant of T or any other false theory, like we did with U, then T is not actually useful at all. Refuted theories can only be useful via non-refuted theories that make reference to them, not on their own.