BLM is not a single organisation, and I find it extremely frustrating that you're talking as if it is.
There is an organisation called Black Lives Matter or BLM in the USA, which has chapters in other countries. Each one is a separate organisation.
But the BLM matter movement is far, far wider than just those organisations and is not controlled by them.
Further, in relation to BLM(UK), there was both a changearound of peole and a change of name, to a new name with the same initials but not the same words.
Meanwhile, there are various unconnected BLM groups in the UK besides the group that changed its name so that their BLM no longer stands for Black Lives Matter.
I wonder whether people did the same with suffragettes (and suffragistes etc.). Probably.
Or the difficulty people have between liberals as in members of Libdems (or equivalently named parties in other anglophone countries), socially liberal people, economically liberal people, and politically liberal people, all of which overlap like a venn diagram, but no two of which are quite the same. Ditto communists - those who believe in pure communism, or those who support distorted communism of the sort that is now associated with the word communism, but is very, very different?
As for disabled people aligning our problems with others, I don't feel that campaigning for one sort of equality means that I can't campaign for another sort. I don't feel I need a stronger argument for doing so than "First they came for..."
If I don't stand alongside other ethnic minorities such as people of colour, or East Asian people currently victims of some rather nasty attacks, some verbal and some very violent, usually with some reference to Chinese virus vel sim., or any other grouping, why would they stand alongside me as our government goes on the attack? I'm getting more and more frightened with so few people speaking up as Priti Patel gets vicious legislation through parliament.
As for who campaigns are aimed at, it depends on the campaign, both in terms of overall campaigns/groupings and in terms of specific campaigns/events. I don't get the reference to 'mainstream'.
As for all racist and all homophobic, well, no, we're not all racist and homophobic.