Why did Jesus promote theophagy, and did other Jewish people at the time believe in it?

"Jesus," or rather the actual historical man (or men) upon whom the legendary figure of Jesus was based, may never have said anything of the sort. It's certainly not a concept found in traditional Judaism. But the practice of killing and eating a representative - either animal or human - of a "god" is widespread in "paganism" - and it's notorious that the Christian religion is largely constructed from elements of earlier pagan mystery cults. See Dionysus, who was often represented as a goat, ripped to shreds and devoured raw at the Bacchanalia. Much of the symbolism of the Eucharist is directly derived from the worship of Dionysus.

Or a sacrament. Jesus also said that the Words He spoke were Spirit and that one needed to understand them spiritually or in others words believe what He taught by faith. He of course was speaking of His death, Resurrection, and ascension and that one must partake of that to live. We feed spiritually on Jesus even as we feed on food for our physical bodies.

We speak in the same terms so it should not seem strange. When someone tells another a lie and then they say that they "swallowed it whole" we mean they believed the entire lie. Or we say they took the hook, line and sinker--etc