I'm leaning Sith right now if only because ~~Jason~~ Qimir started in on "peace is a lie" for that split-second.

I do think "philosophical beef" is probably the right answer, though. Because if some random orphan, taking Forcin' classes part-time on a beach with some shady-lookin' drifter, can not only match but surpass a Jedi master in a Force-off... That's gotta do some serious damage to the Order's reputation, right? Especially if it comes out that the Jedi specifically failed to train her identical twin? Imagine you're one of the parents whose child went off with the Jedi because it was the "best way to realize their cosmic destiny", never to see them again because the order drills "attachments are spiritual poison" into their heads from childhood, and suddenly you find out that, basically, homeschooling was an option the whole time? Now you start thinking maybe those "enlightened peacekeepers" training your kid to "harness their spiritual potential" might just be... kidnapping children and indoctrinating them into their weird death-cult to raise a galaxy-spanning theocratic paramilitary with a forced monopoly on psychic phenomena, against which the common denizen has no defense — who are, incidentally, hoarding all knowledge pertaining to their Order's history, philosophy, training, and religious practice on inscrutable magic prisms that only they know how to open and read, which they keep in a locked vault in the back of their heavily-guarded stronghold. Trust in the Jedi erodes, maybe folks are a little more hesitant to send the youngsters off to Force Camp, and the Jedi Order either stagnates without new recruits or they start taking drastic measures which will, of course, only erode the public trust more, rinse and repeat until the Jedi go exctinct, and the Sith kick back with a couplea space brewskis and have a grand ol' chuckle. "Kills the dream" indeed.