Why do scientists have such trouble with the theory of time? (Einstein turned it into an aspect of space; Godel seems to have denied its existence, except as a kind of subjective illusion.)

Maybe because time is not a thing.

Time has no tangible properties in itself - it's only manifest in its effects. It's not even a mathematical object - it can't be quantified (except in "relative" terms - & the theory of relativity has nothing to say about time as we know it subjectively).

Godel's interpolation of the General Theory of Relativity hypothesizes "closed timelike circles" open to "time travel" (though it's not strictly time travel, because there is no time).

Fiction (& sometimes poetry) - & not just science fiction - provides forms of virtual time travel. As we write, we relive. A la Recherche du Temps Perdu.