We know they didn't wander in the desert for forty years as the trip only takes a few weeks (at a slow pace). If that didn't happen then the whole rest of the OT is fiction (and reads like it). Psalms, one of the longest books and basis for others, is a series of poems.

I personally suspect that the Exodus did not take place, but there certainly was an exodus of a kind. There was, and we know this because there are historic records, an almost continuing exodus all the time. Since the semi-desert lands of the Sinai to the East of Egypt, were the almost inevitable destination for all escaping criminals and runaway slaves, coming out of Egypt, there would have been an almost endless flow of such people all of the time. It was therefore inevitable that the people, living in the deserts to the east would have come, in the end, to believe that everyone had once come out of Egypt, and would eventually forget that there were originally some natives who had always lived there.

Then as always happens, in myths and story telling, most of the real events would fade from memory, and only a single notable event, perhaps not even the biggest, centered on a mythical hero figure, who may have been made up out of the stories of several people, would remain.