Not kleenex, no, but open source as a term exists in the same way that escalator does. If the OSI was to change their definition, they would be wrong. The definition has transcended them. I honestly have a hard time understanding how people don't agree with that.
But due to how trademarks work, they would have had to then aggressively protect that trademark with lawyers and stuff and I have a hard time imagining that working out all that well with a lot of us open source software enthusiasts and companies.
Simply put, I think if the name “open source” was a trademark, the open source community would still exist, and businesses would produce open source software as they do today too, but a lot of us would be using some other name for this instead of referring to it as “open source”, so that we’d avoid any potential problems with the trademark of open source.
It wouldn't be such a big deal. As the current situation with VVVVV shows, we already kind of have our own little informal community-driven "trademark enforcement", and it works.
I used escalator specifically because a lot of people don't realize it originated as a trademarked term for the Otis Elevator Company's moving staircase. Kleenex is still well known as a brand name for tissues, even if it is used generically.