Equally, if employee Y is producing the same quality of work as employee X but trying to charge more money, does employee Y not deserve to be out-competed by employee X?
Employment is a two-sided market too.
Nobody would advocate for a minimum price for a packet of crisps just to save the poor crisp companies from earning too little per packet. It is obvious that if crisps were too expensive, people would stop buying them. The same applies for workers.
Minimum wage is just saying "if you can't produce at least $X of value per hour, you're not allowed to work at all". That's not fair.
What you are arguing for is a welfare state.
As an example, modern Germany had a long tradition of welfare state combined with no minimum wage.
Welfare state as in? Having social benefits and minimum wage is not mutually exclusive.
Furthermore according to Wikipedia [1] the minimum wage in Germany is 8.84 EUR. You might say, "that's Wikipedia, that's no reliable source!" No sweat, go click the link first. For this specific information (minimum wage in Germany) they provide links to an English AND German source.
[1] https://en.wikipedia.org/wiki/List_of_minimum_wages_by_count...