Stigma. It simply isn't done that much. Even Jeff's list proves that...30,000 golf courses in the world and just his list, which could probably be expanded to what...maybe a few hundred on regulation courses?
Arizona Grand is my only regulation course with a finishing par-3. And even this was once a short four until neighboring land was appropriated.
I think there is a general feeling that a demanding hole requiring several tests of golf — perhaps accuracy, carry and some length — is called for at the end of a match. I used to like par-5s, and still do, because they tend to somewhat equalize play if the length is "correct" and the risk-reward ample. Of course this equalization is also a trait of the par-3...but it seems not quite as demanding of multiple skills. Whereas the par-4 or par-5 requires a banger tee shot, a long club from through-the-green and (usually) an approach or short testy shot...plus the putts and green surround shots that may be at play.
BTW - A par-3 opening (first) hole is the best way to self regulate pace-of-play. A par-3 first hole simply meters the pace without much need to a starter. But we rarely see this at all.