Yes, but A: goals and effects are two very different things and B: it was created in an increasingly foreign world and every year it is less obvious that the original logic holds. (I mean that in a very general sense, not in an immediate "discard minimum wage and return to a libertarian paradise" sense; as robots become more and more capable and really start eating into the low wage jobs we may have to consider things like guaranteed minimum incomes or other radical proposals.)
"Employers as a rule extract a multiple of value from their employees time, not just a little bit over."
In my opinion, not if you do a full accounting. Profit margins for established, successful companies in the vast majority of industries tend to run in the single digits percentages, bursting up to 10% or the low teens for some rare cases. And then there's the tech industry, which is another story altogether. It may be true in some abstract sense that companies extract multiples of value over what they pay but it isn't true in a useful sense.
Generally speaking, I deny that there's a huge amount of room for people to cut minimum wages, because employers do not make radical profits on their employees. The job market isn't perfectly efficient, but it's more efficient than that. A person/task that can't justify minimum wage in the modern economy is a person not hired and a position not created, not a person generating 2$/hr of value but forcibly paid minimum wage.
Now, if the economy was grotesquely inefficient and the average company made 200% profit and the only way to pry money out of their hands was with a minimum wage, I'd agree, but that's not the world we live in. It's certainly a world we hear about a lot in propaganda but it doesn't correspond to reality.