If the history was primarily a matter of brand recognition, as you have suggested, then a much higher fraction of the studio players would have used alternate brands. The studio players are ruthless about choosing tone first, brand second. The fact is that there simply isn't a presence of Gibson basses in mainstream recorded product for literally decades after Gibson and others had introduced their basses. Brand recognition simply does not explain the historical record.No, all it says is they were first to market with a newish type of product and to gain acceptance, and they became the generic name of that product type. It's no different to tissues being commonly called a Kleenex, or a vacuum cleaner being called a Hoover, or...
If Gibson had gotten there a few years before Leo did, then the generic term would likely be Gibson bass today.
Another way to put it is: watch Woodstock and do the math. And Woodstock doesn't even include the handful of studio players who created the vast majority of the recorded literature.
Aside from all of that, it is perfectly ok and appropriate for a player to choose otherwise. Most of my handful of instruments are non-Fender. And, there is an undeniable brand recognition thing with Fender; which is why I choose to take Fender on the headstock (usually a parts bass I constructed) to auditions. But, that brand recognition is grounded in performance characteristics, not just name recognition. There is a difference.
TL;DR, carry on.