I'm not really sure what you're trying to accomplish. If you want to eliminate subtours upfront, you can add the constraints right away and then there are no sub-tours (as you observed). Like already mentioned by Daniel Junglas, the problem turns from MILP to MIQCP if the constraints were added in this form. If you are linearizing them, then there are more than $|S|$ constraints, which still explains the slow down. The problem with MTZ-like constraints is, that (to the best of my knowledge) you cannot add a fraction of those, it's either all or nothing, so it doesn't really make sense to me, to add those later on as a lazy constraint.
Considering the fractional solutions: I don't know how your linearized equations exactly look like, but as far as for the MTZ: the relaxations produce very poor bounds so this is also something to be aware of.
Also bear in mind, that your formulation (as well as the MTZ) only work on ATSP and if you got a symmetrical TSP and are formulating it still as ATSP you are introducing a lot of symmetries, which is never good.
What exactly are you trying to accomplish and is it worth it? As far as i know, solving an integer model and adding "standard" SECs as lazy constraints in the callback already gets you pretty far with the current solvers. Could you elaborate more on your goals?
