@sand15 I consider it a weakness that simplify is not trying to obtain a normal form (in terms of radicals) for the whole expression as one kind of preliminary step. That seems a natural thing to attempt here, since the expression contains radicals and is of modest size, complexity, and degree.
I did not explicitly state that I consider it a bug because it does not try expand on the whole expression. Perhaps it ought to call expand on the whole expression as one avenue of attack -- I leave that for someone else to figure out. But I do think that it could try radnormal or evala.
The simplify command must involve some heuristics, on practical grounds. I think that this example merits at least consideration by whoever is most expert in its workings.
Have I understood your question properly?