Word for a "false idea with a pragmatic outcome better than that of the truth"?

Solution 1:

A recent coinage for the concept you raise is epistemic innocence. This term comes from the interdisciplinary Project PERFECT, which investigates whether false beliefs (especially delusions, distorted memories, and beliefs that don't reflect social realities) could have positive effects. When inaccurate beliefs do convey a benefit, they may possess epistemic innocence (Philosophy Now):

Similarily, Bortolotti argues that inaccurate or imperfect cognitions, for example delusions or factual misrepresentations, can be epistemically innocent if:

(a) They provide ‘epistemic benefit’ – meaning, they’re beliefs that can help us.

(b) There is no available alternative that would confer the same benefit without higher cost in terms of knowledge or beliefs. (‘Epistemic’ means ‘referring to beliefs or knowledge’.)

The trickiness with a usage like this is that it's jargon (even if it's on Wikipedia), and very recent jargon at that; it is likely that even philosophers and psychiatrists will need the phrase to be defined for them. In using the term, you may be signalling that you agree with the theories of a particular group. So keep that in mind if you decide to appropriate the phrase.