"FAXing over VoIP networks doesn't work."

Too much of a generalized statement to be correct.
I say
FAXing using a VoIP ATA that tries to pass high speed FAX analog works sometime, with some VoIP providers, and with some degree of congestion on the ISP/Internet. better success can be had by changing your FAX machine to use a low speed, say, 4800 baud.

VoIP on private networks such as frame relay often works.

Almost all FAX attempts I made using xxx VoIP service succeeded, but not so others.

It seems that if you have an ATA that supports the T38 standard, analog FAX can work just fine - IF your VoIP provider does end to end support of T38. Few do.

Therefore
FAXing over [low cost commodity] VoIP networks [with cheap ATAs] doesn't work.