Setting up a Mathematical Institute of Refereeing?
I'm an assistant professor of mathematics and while I regularly referee papers (in fact, I have a few papers I'm supposed to be refereeing right now), I have to say that joining a Mathematical Institute of Refereeing sounds awful. It's right down there with joining a Mathematical Institute of Departmental Meetings. 

Perhaps it's just me, but the aspects of my job that I love the most are being able to teach mathematics to and mentor talented students, working alone or with collaborators on research projects, and writing up / publishing papers. Although I definitely get something out of refereeing papers (and seem to referee ten or so a year), this is mainly something I do in order to be a "good mathematical citizen". That is, I expect that when I submit a paper it will be competently refereed, so it's only fair that I referee the papers of others when I'm qualified and able to do so. There's no way I would give up all of the aspects of my job that I love in order to focus full time on refereeing. (If such a thing were even possible. I'm not sure how my college would react if I told them that I'm going to be refereeing papers full time for a while but will be back in time for the Fall 2024 semester.) At the end of the day I think what it boils down to is this. I also enjoy doing yard work. But I'd never consider quitting research and teaching in order to focus on it full time.