Goods and services do not have an intrinsic value; their value is determined by scarcity and demand. If dish washers are not scarce, their wages are low and they should be low. The idea that labor or products have some intrinsic, absolute value independent of scarcity or demand is common in fascist and Marxist economics, and it simply doesn't work in practice.
I agree with everything here (when I said "actual value" I meant that colloquially as value to the consumer", sorry about the ambiguity) except "...is common is fascist and Marxist economics..." It always seems to be the pro-capitalists that talk about value as independent from demand. In fact, you do it immediately in the next sentence:
the price is determined by the cost of the inputs plus the value the restaurant adds" [emphasis added]
No, the price is determined as the equilibrium between what the customers are willing to pay and what the vendor is willing to accept. The cost puts a floor on what is profitable to the vendor, but that's it.
If one of the inputs (e.g., clean dishes) becomes more expensive, then the customer will just pay for it.
This presumes that the customer is willing to pay more. The value to the customer of the end product hasn't changed so why would this be true? But, yes, if the amount that the vendor is willing to accept changes then a new equilibrium price will be found and that new price may be higher. It could also be lower (especially if the new minimum wage puts more people on the border line of willingness to pay, in this case price discrimination is the key).
Anyway, all I was pointing out is that there are situations at the minimum wage level where the value to the business of getting the job done is higher than the current cost to the business of getting that job done. While businesses may complain about it, they'll pay the higher wage for those jobs.
(Regarding substitution: I assume that most restaurants already have industrial dishwashers, but those dishwashers don't load themselves.)