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.
That's an oversimplification, since it ignores what is paid to people who produce more than $X of value per hour.
Minimum wage says something closer to, "If you can produce at least $X+Y value per hour, you will be paid at least $X", where X is the minimum wage, and Y is the level of profit that a company needs to operate.
Assuming no upward pressure on X, a company will prefer to maximise Y and minimise X. Minimum wage laws are one way to apply upward pressure. Competition over constrained supply is another, but only applies if there actually is a supply constraint.
It's more efficient and humane than basically telling them they're worthless and they should just stay unemployed.