In a toy economy that's definitely true, but the empirical results are mixed in the real world, where there are lots of complicated dynamics.
(Yes, I know that often businesses miscalculate the value of an employee, and it may have other sources of profit that cover the losses, etc., but it doesn't change the reality that the profit motive will squeeze such things out.)
Thinks are complicated. All the dot-to-dot statements are somewhat true, but they're all going on at the same time. Higher minimum wage means more spendable cash, which means more small businesses see an uptick in sales, so more profit. See! I can do it too.
Deadweight losses are the inevitable leftovers of decreased efficiency after they have been drawn.