"Given that the International Phonetic Association (IPA) started in France
(in 1886), this seemed rather odd." Richard WORDINGHAM
The IPA was founded by French scholars so they took the non-nasalized vowels
of French as the cardinal ones. Had it been founded by Italian scholars, I
suppose they would have taken the Italian vowels as cardinal ones. Ditto
with Spanish or Portuguese.
Whether you take any of these romance languages, there is no schwa. The
concept of schwa was borrowed from Hebrew, and developped for Germanic
languages as a mid-central vowel, then applied to other languages.
That the IPA symbol for schwa should be used for the "e muet" in French is a
mere convention that, to me, does not reflect any phonetic reality.