Theoretical physicists are, on the whole, not a silly bunch and there do exist various "mathematically identical" ways to look at quantum field theory. But by the very definition of "mathematically identical" one can find the path integrals somewhere in these alternative descriptions anyway.

Contrary to your stated desire, I actually think that the path integral formulation is in some sense the most "intuitive" and "local". I apologize for not taking the time to explain that here, but let me instead try to convince you that "locality" is not at all obvious in quantum field theory.

One intuitively non-local process is the following. If I start with a state with one electron at the origin at time 0, what is the probability that I measure an electron one lightyear away one minute later? Naively this probability is zero, because the electron that you started with cannot move faster than light. But the actual (and measured, in a less idealized setting) probability is not zero, since vacuum fluctuations can create an electron exactly one lightyear away and you end up measuring that one. That, a physicist would say, is perfectly cromulent with locality of the underlying theory.

I hope this gives a glimpse of how subtle causality (and hence locality) really is, and why at least some believe that path integrals may actually be the most "intuitive" description available.