The empirical evidence on a relationship between minimum wages and economy-wide unemployment is extremely mixed. How you describe it is how things work in toy economic models, but any model with a hope of being realistic would at least have to account for the fact that employment demand isn't external and static. Other complexities include how it interacts with automation: in some jobs, artificially raising the cost of low-end manual labor just moves forward a few years the point at which it becomes cheaper to automate the job, which may be good or bad for overall employment.
