You're conflating two different sets of people.
The overall problem is nepotism, or even more generally, a lack of upward mobility. If you're poor it's a long road to the top. Most of the slots are filled by nepotism or otherwise having rich parents and only a minority are filled through merit.
The ideal solution here is to improve upward mobility, but neither party really does that, because to do that you have to fight entrenched incumbents. To lower poverty you need to lower the cost of living and therefore housing and healthcare costs, but the existing property owners and healthcare companies will fight you. To create opportunities you need to reduce regulatory capture so that small businesses can better compete with larger ones, but the existing incumbents will fight you. So these problems persist because neither party solves them.
Then, because more black people are poor and the bipartisan consensus is that if you're poor you're screwed, there are proportionally fewer black people at the top. Now consider what happens if you propose DEI as a solution to this. All of the nepotism still happens, but now the merit slots get converted into satisfying the race quota. Now if you're poor and white you're completely locked out, because the "white people" slots are all filled by nepotism and the remaining slots are used to satisfy the race quota.
The result is that poor white people are completely screwed by DEI, they understand this, and because the proponents of DEI are Democrats they then vote for Republicans. Then the Republicans oppose DEI because they're actually representing their constituents who at least want their chance at the limited number of merit slots instead of being completely locked out.
If the Republican elites then engage in nepotism and fail to improve upward mobility, that isn't good, but it's only a distinction between the parties if the Democrats would have done better in that regard, which they haven't, and they're plausibly even worse in terms of increasing regulatory burdens that prevent people with limited means from starting a small business.