So Jesus (or Matthew, writing fanfiction about Jesus) was the libertarian who gets mad at raw milk not being sold in stores instead of actually harmful regulations? Jesus makes no attempt to defend "your disciples don't wash their hands" with "they do, just with a different way."
But should we expect Jesus to have defended His disciples in that way? Only if your reading is right, and the Pharisees were concerned with sanitation rather than ceremony. But that seems to read a 21st-century conception of handwashing into the 1st-century text.
On the other hand, Jesus' response provides useful context in understanding the Pharisees' challenge, and it addresses spiritual purity, not health. The Pharisees' very question suggests the same, because they frame it in terms of "the traditions of the elders," not "health," as you misparaphrase it. (Note that the parallel passage in Mark may further support this reading, since Mark explains, as the
ESV has it, that "the Pharisees and all the Jews do not eat unless they wash their hands
properly, holding to the tradition of the elders." The Greek there, interpreted literally, reads, "unless they wash the hands with a fist," which the expert editors of the ESV suggest "probably indicat[es] a kind of ceremonial washing." The
NIV translates it as such explicitly: "unless they give their hands a ceremonial washing.")
