Today the majority of free software that I use seems to be Apache/BSD/MIT - and GPL for Linux - but back in the 90s everyone seemed to make up their own “free” license, and there was a lot of confusion around GPL. My recollection is that OSI was trying to create some clarity around which licenses really were “free”, and which weren’t.
I was just a passive observer, I was not involved in any way, I wasn’t a fan of some of the players, and I even thought calling it OSI was pretentious - but nevertheless they made a dent in the universe by popularising the term “open source”. OSI was far more than a rebranding because it enabled us to speak about GPL - which imposes conditions on use - and free licenses like BSD in the same breath, while excluding what we now call “source available” licenses.
So despite my misgivings, I think the OSI deserves credit. They stood up and did something meaningful, and helped move the entire software industry forward.
Because of this, I think calling it a rebranding is totally unjustified and unfair.