But if you set your minimum wage to $20, then this becomes part of the cost function for the producers (assuming everyone is fully trained and there are no shortcuts to take--the LTV default--by adding it to the fixed costs). And producers are free to try to find other ways to reduce the costs (e.g. by paying more to higher-productivity employees that more than make up the difference, which LTV has trouble grappling with). MTV has no trouble with incorporating labor costs; it will just tell you that some things aren't worth making because nobody values them enough.

(Note--you said your example took into account "the underlying social relations", but it didn't. There weren't any social relations in the example, save a statement of there being a "socially necessary wage rate [of] $20 per hour".)

LTV just doesn't have any advantage save as a moral argument, and it's not a very good moral argument because either the moral impetus comes from fiat (which is no argument at all) or because it's a good model of what we actually value (but in many cases it isn't).