No, Beethoven could not have written Tambourine Man. Beethoven was a man of his time, and music was bound by strict rules that controlled what was acceptable in terms of harmonies, and physically bound in what instrumentation he could write. Beethoven pushed out the rules a little bit but he was not anywhere near where Tambourine Man is; the very concept of the modern I-V-IV progression would be enough to give musicians of the time the vapors.
Part of the reason the classical musicians were so voluminous in their output is that their rules were so constraining that once you wrote down the melody, half the work was already done for you; there were a very limited number of choices left to you. I enjoy Mozart a lot, but at the same time listening to him with modern ears it is obvious that he has a very strong signature style, and that he could simply crank out the tunes using that style in a manner not entirely dissimilar from how one cranks out autotuned pop hits using standard harmonies and singers that can't actually sing. Mozart's results were better but there's more method overlap than you might think.
I think when people say that all the good composers are in the past, they are really just socially signalling. It's not actually true. Plenty of spectacularly good people are around today, and like I said before, if they seem less unique it's because the top end has become crowded, not depopulated.
Despite having no physical limitations of relevance, Mozart would never have written even The Flight of the Valkaries or much of anything else Wagner did, let alone Debussy, or Gershwin. All the classical musicians, geniuses though they were, were also very very confined in their style. They did not exhaust the space of good music; they only took the first hill of what we'd call modern music. It is a disservice to the many that have come after them to pretend they were uniquely virtuous; they were merely uniquely first.