Someone did: PalmSource (the software spinoff from Palm). It was to be the basis for their Palm OS “Cobalt”, which was eventually abandoned.

Or by “someone else”, did you mean someone other than PalmSource?

Let’s be honest, though: most new OSes don’t enjoy much success or longevity. The only successful new arrival I can remember recently was Android, and it had the backing of Google, and was aimed at a relatively new market. It also borrowed heavily from established tech: Linux kernel, JDK and JVM.

I doubt there’s more than a handful companies that could have turned BeOS into something big, and of those, Apple was probably their best bet.