I think if you asked a BLM supporter whether they thought "all lives mattered" they would say, 'of course they do, but it's mostly unarmed black people getting shot by police, black people doing time -and virtual slave labour- in US jails, and black people making up most of the numbers for social deprivation, so if you really want to show that you think "all lives matter", then that gives you something specific to think about. You might also think how it was the slogan "black lives matter" that led you to pay any attention to this in the first place", and how scoffing at that slogan with a knee-jerk "all lives matter" reveals a preference for ignoring the consequences of racial prejudice, however deadly.'
Of course they might just say "F*ck you, wh*tey", but then there is rather a lot to be cross about. (And to put things into perspective Just think how cross we get about Kwis, or copper bullets, or people who think the latter are called "heads", or who can't spell "muzzle brake" correctly. [OK, that last one may be just me.]) But, honestly, black people in the USA have a lot to be ****ed off about, and I reckon it's an un-fairly bumpy ride for non-whites in the UK, too.
My take on this is that if we can fix the problems black people face we will be making our society better for everyone: in short, if we make sure black lives matter we will get much closer to making sure all lives matter and everyone gets the chance to use their days on earth well.
Think of it like conservation on game shoots: if your grey partridges are thriving, you can be pretty sure the rest of the habitat is both diverse and doing well. This is the opposite of the Kwis b*llocks, which favours apex predators while not giving a stuff about what their presence or numbers mean for the wider ecosystem.