Besides “the” flesh they were just talking about was His flesh.
Christ often spoke mystically. Spiritual experience (always referencing the bible, as the “high minded” did, to look for whether, and if so how, it fits) is necessary to figure it out. This could even be a dual-meaning dictum like many in Scripture are. God is not averse to punning... what would you expect out of a joyful God?
I lean on the explanation by context. Passover seder, and the same words “this is” would be used over all the symbolic foods. The Passover Haggaddah as Jews have it goes back almost to gospel days and supports this strongly. Making it stand out as special here is something that happens when we lose sight of context.
They why didn't he same "My flesh profitith..."?