You are absolutely correct. You could download Unix from the University of California, Berkeley and install that operating system on your own computer for free. Of course, it would take you hours to download the software, and Unix was not easy to configure for your specific computer.
Linux came along and copied that free UCB software onto CDs and sold it as an easy to install package for PC computers.
The rest is history.
Why was UNIX free from the University of California? It's development was funded with government funds. The majority of the operating system functions would be classified as "shareware" today and were written by private individuals.
The UNIX operating system that we see today, was a group effort. If some smart individual created a new program like "ls" to give a text list of the directories, then it would be incorporated into the basic package. That is why you have such strange sounding and unusual commands such as; ls, grep, ping, and others.
This would be like downloading each and every software package available from www.shareware.com and packaging it as a new operating system.
The basic kernel is what UNIX is all about. It and been, and is, simply outstanding and does it's job very well. Everything else is just shareware and hype.
Today, I sadly shake my head when I listen to this Linux v Microsoft debate.