It may have less of an effect than it is claimed, in some jobs the demand is there regardless of wages, so any increase in cost gets transferred to the consumer. If your janitor gets a 20% increase, it is doubtful he will get sacked because he is needed. But if the same happens to a person packing boxes or working and McDonalds, he might lose a job to overseas or a new automated machine.

If you take it to extreme, it's obvious it's not a myth. Imagine a 100k USD yearly minimum wage. Obviously all unskilled worked wouldn't be able to earn this much for the employed, but in places where you have to have someone do the job, they'd enjoy their new high minimum wage.