Would going to Teacher's College be useful, if my ultimate goal is to become a university math professor?
Spending the next year in Teacher’s College, I’ve been thinking, might actually be a good thing, as a way to set me apart on applications for grad school and for jobs in the future, as well as a way to develop good teaching and other related skills.

If you want to get into a research Ph.D. program in pure or applied math, this will do extremely little to set you apart.  Taking more graduate courses and doing well on the GRE (especially the math subject GRE) would certainly do more than education or curriculum courses.  

The most basic reason for this is that Ph.D. programs are essentially only worried about your ability to complete research in that discipline.  Any life experience or preparation that isn't relevant to that is likely to be ignored.  I suspect education courses are particularly likely to not be helpful, since (at the risk of way over-generalizing), many mathematicians have a pretty low opinion of the way classes in education are taught.  I suspect, for example, actual experience teaching in a school would be more helpful.