I think part of the interest in the "difficult" equations comes from domain experts (i.e. not ML people, not numerican analysis people) who would like to make their life easier by applying ML to a problem they need to solve anyway.
But there are also ML people trying to make themselves useful to other disciplines.
Why don't they test on the heat equation first? I suspect because the existing methods are so good that it's a hard benchmark to beat. And ML people have this unfortunate "beat the benchmark" mentality engrained deeply now in them. ML methods may not be as good as hand-optimised methods on simple problems, but they could be better on very complex problems where hand-tuning is too expensive. But that's not very convincing to the reader, is it? What a conundrum