No, you are making the straw man argument. The argument is not that the exact same people have been holding the exact same jobs with exactly the same pay the whole time. There's a reason you found it so simple to lampoon. That reason is you chose to view the argument as naively as possible.
The argument is that any minimum wage earner is worse off than a minimum wage earner the year prior, and the year prior, and the year prior, etc. By holding the minimum wage fixed, we are making the default decision that a minimum wage earner tomorrow should have less effective purchasing power than a minimum wage earner today.
A quick Google search says that a dollar in the year 2000 is worth half as much as a 1980 dollar. Yet, during that time, federal minimum wage went from $3.10 to $5.15, an increase of only about 66%. That means my parents had an easier time surviving off of minimum wage than I did.
This here is the strawman of the argument. Very few employed continuously for ten years continue to make minimum wage / don't move on to better jobs.