This reminds me of someone who once argued that schools cannot reduce their costs when the number of students that need educating declines, because e.g. you can't stop renting a building just because there are 10% fewer occupants.
Is there a flaw in this sort of logic?
You don't need a larger building when there are less people in it, and you don't need more teachers when there are less students.
The argument you described is very different from mine. Mine is more like "schools say books are worth $10, but the prices of books raise to $11. Nevertheless, schools continue buying books"
If you don't believe this, there's data on the subject of minimum wage increases and percent employment. Look at this comment and my explanation below it:
You don't need fewer, do you? You can't fire a teacher because their class now has 27 people instead of 30 - those 27 people still have to be educated.
Do you think teachers are not essential to schools in the way that minimum wage employees are to restaurants? How does that work?
I'm not debating whether simplistic supply and demand analysis applies to the restaurant workers - clearly we have empirical results that say it does not.
I think that to understand something, you have to have data, but you also need to have a theory that is consistent and leads to further insight.
Indeed, if this doesn't happen (which is fully possible) you get an interesting avenue of inquiry about market inefficiencies.
As an aside, I personally know of a case where a school needed to cut costs, and they did that by increasing the student:teacher ratio. That ratio is a metric that gets attention and consideration; at least in the Australian system it is not some arbitrary number.