WetGeek Can't call this one "solved," but I'd be happy to call it "closed."
I was being misled by mixed memories. It's true that I had streamed videos on Opera on those other machines, but that was while they were still running Windows. When I changed them to Linux, there was no need for a second browser on them, so I did everything with Firefox, which was installed as the default.
I just now changed this laptop from Windows (with Solus on a VM) to Solus on the host, and tried to set it up as before, and that's when I got confused. On this laptop, I'd always used Firefox for some purposes and Opera for others. (I love Opera's "speed dial" feature.) I routinely streamed video on Opera as I described earlier, but that was on Windows.
Checking out the other computers last night--the multimedia server, development workstation, and file server--I was reminded that I had not installed Opera on those when I changed them to Solus. There was no need to, given what they were used for.
I feel better with this mystery now solved, and I'm quite happy to use Firefox for those few occasions when I want to stream a TV program on this laptop. Thanks for helping me try to solve what turned out to be a non-problem.