Terence Tao asked for a non-enumerative proof that a positive proportion of permutations are derangements and got a great answer. Inspired by this, I'd like to ask about another family of permutations.
An interval in the permutation $\pi$ (thought of in one-line notation) is a contiguous sequence of entries which is also contiguous in value. For example, the entries $43$ form an interval in the permutation $514362$. Every permutation of length $n$ has trivial intervals of lengths $0$, $1$, and $n$, and permutations with only these trivial intervals are (sometimes) called simple.
Like fixed points, it can be shown that the number of nontrivial intervals in a random permutation is Poisson distributed (see this paper by Corteel, Louchard, and Pemantle), although unlike fixed points, the expected number of nontrivial intervals in a random permutation is $2$. It follows that (in the limit) the proportion of simple permutations is $1/e^2$.
Is there a "non-enumerative" proof that a positive proportion of permutations are simple?