I understand not being from here you aren't as familiar with these slogans and their context. Black Lives Matter as a movement is a movement against police violence that disproportionately impacts young men who are African Americans. Hispanic men and other minorities are also targeted by police, but to a lesser extent.

https://usafacts.org/articles/what-the-data-shows-about-police-use-of-force-by-race/

So the slogan Black Lives Matter is a specific condemnation of this situation. Black lives are disproportionately fucked up by police violence, they matter, ergo we need to do something about police violence.

All lives matter as a phrase, wasn't used as a slogan or anything until after Black Lives Matter. It exists almost entirely in the context of Black Lives Matter, and is used almost exclusively to advocate against reforming the police and lowering the use of force against Black Americans. Its sole purpose is to try and discredit the motives of people advocating for less police violence in Black communities, by implying that by singling out Black lives is not ethically correct. The police act as though Black lives don't matter, so by responding to that by saying All Lives Matter, all you are doing is trying to paper over the problem.

With those explanations out of the way, I think that Dems should not be saying All Lives Matter. It may work better among young white men, but you do so by ignoring a very real and dangerous aspect of the problem. If the Dems want to reach those young white men, I think the answer isn't to water down the existing slogans, but to add legs of the platform to address separate issues.

Show young men how police corruption fucks their lives too. I remember growing up hearing how much worse ticketing and traffic stops are for young men, so that could be a way to push back. Police will do violence to your black friends, and on top of that they issue tickets to men at almost twice the rate they do to women. If that isn't an in to get young men on board with police reform I don't know what could be.

Please don't take that as me saying women have it easier in traffic stops. Cops are horrible, and when cops are sexual predators they are just as dangerous as other predators but alsoalso have state authority to abuse. There have been stories of cops using police databases to stalk women and more, but that doesn't mean you can't bring up men's issues here too. Black lives matter, women need to be protected from cops abusing authority, and men shouldn't be unfairly burdened by traffic tickets. Dems need to learn to find those parts of the problems and speak to them because all three problems can be tackled together.