For your example, I think the domain is x \ge y \ge 0, which can be transformed to (y, z) \ge 0 with x = y + z.
But yes, simplexes (and generally polyhedra) are tricky to transform to a box because you get a singularity at one or more vertices. If you can exclude these vertices (eg if function is continuous, just evaluate it at all vertices and do the transformation so that the singularities are at the ones which cannot be optimal).
Admittedly, these transformations do look a bit tedious and it would be nice to have a method that is clean and pre-packaged and handles everything automatically, but since there isn’t one, it is generally worthwhile to think about transformations.