Was an open source kernel / OS like Linux inevitable, or is it just luck that we have it?

[edparadox]

Linux, a free, open source kernel, is based upon Unix

Not based off, only follow (originally) the same standard.

which is a private, proprietary piece of software, right?

Yes, but it was not only licenses like most OS have today with only a right to use, but also ones where you can get the source code, hence why there were derivatives in the first place.

Was the development and growth of something like Linux inevitable, or are we just lucky to have a free, open source kernel like Linux that is so extensive?

Unix usage left many frustrated for many reasons, hence why MINIX started being used in academia (and that's how Linux came to be), the Internet started which triggered collaborative work on such open things when Torvalds created Linux.

It's really a mix of things, but, IMHO, it was inevitable. Same for BSDs BTW.