Is the Yang-Mills existence and mass gap (Millenium Prize problem) essentially what is required?
Or are there more problems in putting QFT on strong mathematical foundations?
For example, the exsitence of measure in doing functional integration in quantum field theory and string theory.
I think that quantum field theory is on a strong mathematical foundations as long as the mathematics behind it, I mean the mathematics on which it relies, is itself on the strong mathematical foundations and if that mathematics is on a strong logical foundations.
So what you practically wanted to ask is will the solution of the Yang-Mills millenium prize problem put quantum field theory on a strong physical foundations, I mean, will that solution complete quantum field theory in any of the meaningful definitions of completeness of physical theory.
But that is rather a strong philosophical issue because what mathematics do in physics is that it serves as a means of describing the phenomena which occur in the world, and only describing, the question of whether physical theory really explains something or not is not (for me) the problem of mathematics but the problem of interpreting the phenomena which occur in a natural world in a good way, and in a way that is in accordance with the experimental data.
So, basically, what you are asking is, in some sense, related to one of the Hilbert´s problems which, roughly stated, is:
Can we build physics so that it is based on a strong axiomatic system and in a way that such an axiomatic system can be described by the help of mathematics?
But that one, of course, is not solved, and I do not see the way in which it could be solved.