There is a difference in trying to simply proclaim "Black Lives Matter" and "marching under the BLM flag". You can be a part of a movement without being part of a politically-driven organization who hijacks the lingo of that movement. If David Duke posted "ALL LIVES MATTER" on social media, it doesn't automatically make you a KKK member if you say "All Lives Matter".
By your logic, everyone out there who would claim to stand with "Black Lives Matter" must be from the far-left end of the political spectrum. That is not even close to the case.
And like I said, just because some self-proclaimed organization happened to be the first to lay claim to the BLM internet URL, that doesn't mean their views now automatically represent the entirety of the movement.