Applying this to the low end of the labor market is pushed in a lot of Econ 101 classes but it requires so many assumptions to be true that the model only has a passing resemblance to reality.
To use your own logic, you can't say that raising the minimum wage won't also change other variables important to the model but assumed to be constant (maybe it causes companies to outsource, maybe it causes workers to emmigrate, maybe it raises worker productivity) - so if economists are truly constrained to ceteris paribus situations then they have nothing to say about the effects in reality from a minimum wage increase.