This isn't quite right. For one, the probability of being infected is not linear (it's capped at 1). If you do the math, the expected number of people you infect (given you are not infected) is roughly linear-ish.
Correct, it's not. But it is roughly linear in the limit of small numbers of people with a small constant probability of becoming infected per interaction. (This assumption becomes problematic when you see "clustering" of highly social people with other highly social people.)
To be specific, if P is the probability of becoming infected when interacting with a single person, then the probability of becoming infected after interacting with N people is 1-(1-P)^N = NP - O(N^2 P^2). It's easy to see that the limiting infection probability is 1 in this simplified model, and that if N*P < 1 you're looking at close-to-linear growth.
