Food deserts are a methodologically fraught concept. The inclusion criteria are basically arbitrary; vary the distance or income thresholds by modest amounts, and the maps change drastically. It also buckets everyone living in a census tract to the geometric center of that tract, so that people living five minutes walk from a grocery store still count as living in a food desert. On top of that, they don't account for public (or private) transportation, using straight line distance as a proxy for time traveled.

The store-classification criteria also tends to only count supermarkets and large grocery chains, artificially classifying neighborhoods with local, well-stocked stores as food deserts.

Supposing those methodological problems were resolved, proximity to a grocery store still only accounts for 10% of the variance in nutritious food consumption between groups. The other 90% is driven by buyer demand, as is shown in the cases where different demographics shop at the same store[1].

For the small group for which access is truly limited (further reduced to the 10% of those for which their purchasing decisions would actually change), other solutions — such as reducing grocery store "shrinkage" through better policing, therefore increasing grocery store availability in areas where elevated crime otherwise renders them economically unviable — are noticeably under-discussed in favor of the heavy handed solutions of the type you raised here.

All of which exemplifies the typical failure mode inherent to many "socialist" policies: 1. Misidentify a problem, or solution. 2. Increase government control/regulation. 3. Repeat — indefinitely, as the problem hasn't been fixed — forever tightening the ratchet of government control.

[1] https://www.nber.org/papers/w24094