Voting for Democrats is just in their best interests. Democrats have captured the interests and organizations minorities care about. Sure, it's reassuring for Republicans to tell themselves that "Democrats are the real racists" and "welfare keeps blacks on the Democratic plantation". But, actually, black people like welfare policies, and Democrats support them when Republicans don't. Likewise, maybe it's irrational for Asians to vote for the affirmative action party, but they also support the Democrats' economic policies, which makes the contest no contest.
I think a big part of the answer boils down to "community organizing". Yeah, it sounds a bit like a punchline about two-bit politicians from Chicago with nothing on their resume. But why should minorities vote for Republicans, why would we expect them to vote for Republicans? They live in Democrat neighborhoods, so everyone who goes into politics to represent those neighbors is a Democrat, so Democrats have a monopoly on speaking to their concerns. Notice how whenever a black athlete decides to get educated about politics he becomes a Democrat. It's not because he's been brainwashed, it's not because his belief that Republicans are racist is his number one issue. (Even if he does believe that.) No, such people become Democrats because that's who's addressing their issues, that's who's educating them about politics, that's who's trying to speak for them. Republicans would probably fail if they went around to black neighborhoods explaining how tax cuts would help the ghetto -- but have they even tried?
This is part of what makes the Republican party so feckless. (Since I'm a Republican I feel free to criticize.) They very often have no real sense of what the people want. They pander, and most minorities aren't wrong for feeling like they're being pandered to. Even with their own base, they're inconsistent on delivering on the issues people care about. They don't stop illegal immigration, they pander about how We Don't Hate Immigrants, They Have to be Legal! They don't have a plan to increase wages, they have an ideological economic plan to which they've attached a marketing campaign that includes slogans about increasing wages. They don't have a plan to help minorities, they have a tax cut or a policy change to which they've attached a slogan about helping minorities.
In this case, I think the real question isn't "Why don't more people vote for Republicans," it's more like "Why does anybody vote for Republicans at all?" I think this is one of the key questions that gets at the core of American politics today. I know many leftists and liberals are unhappy with the Democratic party, and there's a growing sense that it just represents a vague centrist consensus that doesn't really speak for them. But it's no contest, Republican voters have felt this for years, and voted for Trump in large part because they felt (we felt, I felt, and still feel) that he at last actually wanted to do something.
A lot of politics, I think, is obscured by the emotional rhetoric that we claim to believe in. Like the concept of racism -- which is used by a cudgel against Republicans by Democrats who are often just as racist in their own ways. We can't really have a rational conversation about it. But many times, if we moved to the rational issues, we'd find that people are mostly voting in their own best interests after all. So you'll find some dumb twitter personality complaining about how dumb and ignorant rural voters are and how city people are so much better -- but even if this is a dumb argument, as a city voter it's probably in his best interest to keep voting Democrat after all. Everything else is just noise. Likewise, a lot of the loudest political shouting comes from the few people who aren't really voting in their best interests: "I used to be a Democrat, but then I got redpilled!" "I used to be a proud Republican, but I can't stand how racist they've become!"