It is interesting that you find unsigned integers more intuitive. My experience (also with students, but also analysis of CVE give plenty of evidence) is that the opposite is true: signed integers in C are a model of integers which have a nice mathematical structure which people learn in elementary school. Yes, this breaks down on overflow, but for this you have to reach very high numbers and there is very good tooling to debug this. In contrast, unsigned integers in C are modulo arithmetic which people learn at university, if at all, and get wrong all the time, and the errors are mostly subtle and very difficult to find automatically.

You are right that often you need to constrain an integer to be non-negative or positive, but usually not during arithmetic, but at certain points in the logic of a program. But then in my experience it is better expressed as some assertion.