Like others I haven't had the time to read it in-depth, though I have skimmed it. I have some constructive criticism. If your aim is analysis only, it looks pretty good. If your aim is persuasion I think you need to make a persuasive and clear point at the very start before your relatives shutters come down. I would actually begin with something like this:
"To understand why people might oppose Black Lives Matter, it's necessary to make one thing clear up front, one thing that ought to be obvious but somehow isn't. That Black Lives Matter the organization is not the same thing as 'Black lives matter' the sentiment. Nearly everyone agrees with the sentiment. When someone says they oppose Black Lives Matter, they're saying they don't think the movement and the organization align with that sentiment. A lot of people, including a lot of Black people, go further - they think the Black Lives Matter organization and movement are actively harming Black people. Not just through heightening racial tensions at this time, but because one of their chief demands - defunding of the police - is against what Black community leaders have actually been asking for. You only have to look at White protestors shouting down and kicking out Black people who disagree with them to notice a conflict. Once you scratch the surface, you find a LOT of discrepancies between Black Lives Matter the organization and Black lives matter, the sentiment."
The above is a good introduction because it highlights the crux of the issue right away - that the name is not the thing itself. Because it promises evidence of that assertion is coming. And because it doesn't come from some adversarial point of view. Once you have stated your claim - that BLM doesn't help Black people, you can then go on to answering the question of what its goals really are. As written, you've started from the position of assuming your relative already has this question in their mind. They don't.
As you go on you can also throw in some of the numbers involved with BLM when it comes to money and corporate sponsorship of the movement.
Don't be afraid to throw in a little pointed humor as well:
In short, your essay explains more than it does persuades. Sprinkle more topical and simple examples as you go. Do it news article style with a couple of big bold inserts like
Study X found that Black men are actually more likely to be shot in an incident by a Black police officer than a White one.
Okay, that might be going a bit too far, but then again, maybe not. But there's a lot of things like this you can add such as how many of the things asked for were already given - Chauvin was charged almost immediately. Breonna Taylor's death led to an actual change in the rules for No Knock raids yet protestors in Kentucky are still demanding "action" over it.
Finally, I'd drop the "like any good historian" tone entirely. Change "Now that I have established..." to "So you see that..." They're extending you the courtesy of reading your essay, treat them as a relative not an audience. You write like a stereotypical courtroom lawyer producing a chain of evidence when you should write like a relative showing a line of reasoning. Take a less formal tone, it will do wonders.
