Would eliminating DEI improve government procurement?

Yes, by 15%.  From a paper by some middling economists,

Mergers, Cartels, Set-Asides, and Bidding Preferences in Asymmetric Oral Auctions, The Review of Economics and Statistics, 2000, vol. 82, issue 2, 283-290.  

The SBA set-aside program limits auction participation to "small" firms... The set-aside program has numerous analogies in other areas of government contracting in which, for example, bidding is restricted to minority-owned firms (Froeb and McAfee (1988)).

The costs of the SBA program can be simulated by assuming that small, high-cost firms evolve into large, more efficient firms. The expected price change under this hypothetical scenario is computed over all 51 [Forest Service] auctions... . On average, price would increase by 14.8%.