True probabilists have a rather unique way of thinking. It is, if you will allow word-creation, hyper-analytic. This thought pattern seems (anecdotally!) to not be too compatible with algebraic or logical patterns. I'm not talking about basics, of course, but on a high level. I've never met a probabilist who enjoyed or personally valued the theory of modules, for example. I've never met a probabilist who would feel that the model-completeness of algebraically closed fields was super-cool.
If you're not inclined toward such things, then the foundational advantage conferred by NSA is moot. And the intuitive advantages are already exploited without hesitation. In my experience, all probabilists think with NSA ideas by default and without self-conciousness, and without concern about how to "rigorize" the arguments.
To make my point, everyone knows that Brownian Motion is the limit of simple random walks. They don't feel the need to make this rigorous, it is just self evident. That it can be made to be almost trivial using NSA is as interesting as seeing an epsilon-delta proof of continuity. Fine for beginners, but not something for me to spend time on now.
(Edit) Disclaimer: I make no claim to have met a random sample of logicians, algebraists, probabilists, or anyone else. I was at Urbana-Champaign for a number of years, and had classes/seminars with Loeb, Henson, and Burkholder, and am married to a industrial stochastic analyst. I love NSA and find it gorgeous, and I feel the same way about probability (but not stochastic calculus, sorry). I've seen first-hand over almost 2 decades how students and professors react to NSA, but again it was not a random sample.
We all know that essentially every mathematician has a "flavor" or two that they prefer over the others. Some of us are analysts, some algebraists; I love combinatorics, and many others don't give it much respect. All I wanted to point out was that the "flavor" of formal NSA is distinctly different from that of today's probability, while "infinitesimal" thinking is already incorporated. This combination, in my humble opinion, is why NSA has not taken hold in probability. There are of course exceptions, with Ed Perkins being the most notable but not the only.