Saying Black Lives Matter doesn't discount all other lives any more than saying "Save the Rainforest" means burn the redwood forest instead or "Save the Whales" suggest f*ck the dolphins. It calls attention to the way that black lives and black people have been treated as less important than white people. It calls to attention the systemic racism of our justice system. It shines a light on the racism of so many American citizens and it does it in a way that many people, even people who are not racist (or don't like to think of themselves as racist), find uncomfortable.
We should be uncomfortable. When we who would be allies to the oppressed and mistreated step back from language that makes racists uncomfortable, even something as simple as saying all lives matter, when it is incredibly obvious that all lives do not matter equally in our society, we are giving aid and comfort to the racists. We are letting the "mildly" racist feel smugly self justified in their better and more inclusive phrasing.
Don't make racists more comfortable. Make it uncomfortable to be racist. Make it very clear that racism, even "casual" racism or racist "jokes" have no place in society.
Make it scary and uncomfortable to be racist.