That is one of those questions which can only be responded to with the answer "it is purely academic".
I find the possibility of his existence -virtually indistinguishable- from the possibility of his non-existence, on the basis that the amount of physical evidence amenable to present-day analysis is identical in both cases.
That amount is, approximately, nil. To the best of my understanding.
If I recall rightly, the Romans believed that a potential rebellion was afoot and that the leader needed to be not merely executed but any possible post-mortem rallying point obliterated: - tomb, bones, robes, spouse & offspring (if applicable), household, handiwork, letters (if he was even literate).
If it took 40+ years for a 'movement' concerning his life to get underway (and the business of putting two and two together to make things fit with OT prophecy), then their efforts to recover traces could only ever be retro-active. The gospels describe how his body was spirited away and entombed at a location unknown to the Roman occupiers. How very convenient. ;-)
I would argue that the Romans would have patrolled the crucifixion site, knowing how many days it usually took for prisoners to die and that they wouldn't have let this would-be-King's body out of their sight, for as long as it took to complete the task of desecrating and/or incinerating it.
It's grotesque, I know, but the idea is to put on an exhibition for any other aspiring rebel leaders to get the message.