In type theory every definable natural number is a program which evaluates to a concrete finite numeral. You can't get more grounded than that. Of course, there is still a large variety of standard and nonstandard models of type theory as well, but the computational interpretation already corresponds very closely to intuitions about intended (standard) models.
In contrast, the lack of clear computational meaning in classical theories makes it necessary find philosophical justifications, which in turn usually refer to other theories without clear computational meaning. Of course, we can compile classical proofs to programs as well through a variety of transformations, but they tend to be sort of unsatisfying, for example we may get functions with empty domains that we can't actually call, instead of programs evaluating to numerals.
So, Peano arithmetic is just too loose and fuzzy for my taste. It's full of things which aren't numbers, rather statements referring to things which have properties which we think numbers should have. And then we can choose between sticking to first-order logic and leaving non-standard models in, or switching to second-order induction which lets us prove more statements at the cost of completeness and leaning more on ambient set theory.
Simpson seems to be critical of second-order logic because it's set theory in disguise; but to me that kind of dispute is moot because I find any sort of classical logic unsuitable for mathematical foundations.