It shouldn't piss anyone off, and I really don't think it'd even piss Bruce off if he was aliev today. Mixed Martial Arts is really just furthering the idea that he...hrm, not that Bruce conceived (because I'm sure there were plenty of folks mixing and matching martial arts long before he came along), but maybe that he forged a path for into the mainstream. "Use what works" was more important to him than "do things the way I do them," so I really don't think he'd be offended at someone pointing students to an MMA school instead of folks diligently practicing what Bruce did almost forty years ago. Folks looking to learn "Bruce Lee JKD" can go find those sort of schools, people looking to learn what JKD has evolved into can be more than happy at a regular old MMA school.
My own JKD/Escrima instructor studied alongside Dan Inosanto, and has certification, etc, right from Dan. He tends to teach us stuff the way he was taught it, because that's what he's the most comfortable with (so a "classic" school) -- but our assistant instructor has studied an assortment of different martial arts throughout his life (and was a combatives instructor in the USMC for a time), so he's right there to show us different ways to accomplish what our main instructor shows us, little variations on a move, a nasty follow-up, etc, etc...so that overall we're left with the option, with almost everything we learn, and it's up to us what we want to take away with us from class. I feel pretty lucky that one's big on the martial, one's a little more focused on the art, and we get to hear them both.
My own JKD/Escrima instructor studied alongside Dan Inosanto, and has certification, etc, right from Dan. He tends to teach us stuff the way he was taught it, because that's what he's the most comfortable with (so a "classic" school) -- but our assistant instructor has studied an assortment of different martial arts throughout his life (and was a combatives instructor in the USMC for a time), so he's right there to show us different ways to accomplish what our main instructor shows us, little variations on a move, a nasty follow-up, etc, etc...so that overall we're left with the option, with almost everything we learn, and it's up to us what we want to take away with us from class. I feel pretty lucky that one's big on the martial, one's a little more focused on the art, and we get to hear them both.
Bruce did popularize the "toolbox" approach, but he never abandoned his original fighting principles. The thing about the "classic" JKD students is that they still take time to evolve and grow, while retaining the core principles of Bruce's fighting style. Too many "Concepts" guys toss out everything, ape what is popular, and call it JKD. You don't know what you're getting. I know many people who think they're studying Bruce Lee's personal style, when they're really just getting a MMA class with a fancy name. This is why lineage is important! It can tell you if the art you're paying money for is the art you want to study.
IIRC, recently the Bruce Lee estate has taken steps to rein in overuse of the JKD name. The "Classic" group are going to be the only ones allowed to use the name; I have no idea of what'll happen to the others.
Most of my Bruce Lee experience was with the older crowd: Taky Kimura, Jesse Glover, Patrick Strong, James DeMile, among others. Their perspective is older, but they all learned their lessons well, and have evolved to suit the times.