https://screenrant.com/enterprises-mystery-chef-should-have-been-1-of-3-star-trek-characters/
The three alternate IDs they suggest for Chef are:
1) Kirk's ancestor (and Kirk himself) - Via time travel, impersonating his own relative
I've heard this before. Heck, they could have had Shatner be the real chef and Riker be the holodeck player - so Riker cosplaying a Kirk relative. For interest's sake, have it be a relative on Kirk's mom's side.
2) Sisko's ancestor
Establishing a long family history of cooking and space exploration. Hmm, for added mystery, have it be a Russell ancestor.
3) Pike's ancestor
Well, we didn't know Pike liked to cook until DIS/SNW, so...
Nah. Reeks too much of 'small universe' syndrome, which Trek already has too much of.
Let Chef just be chef, a somewhat enigmatic figure whose path intersected with Great Men for a short time before vanishing into the mists of history.