If you reduce the supply of labor, while demand for labor stays the same, you increase the price at which existing labor can demand for their services.
You forgot a factor. Unions also demanded shorter work-weeks and days. This opens up two-and-three shift operations, so the demand eventually increases, the supply is the same, but the labor is worth more and more people can be put to work.
Correction(s):
Minimum wage laws and trade union action raises the price of a unit of labour. In economixspeak that
shifts the supply curve left. Unless demand is inelastic, or the demand curve slopes the "wrong way" (not likely) then these developments
reduce demand for labour, which clears at a higher unit price.
Restricting hours is no different from asking for wage increases.
If government legislation is what raises the unit cost of labour then (again in economixspeak) this creates a
deadweight loss, because presumably some employment contracts which would have been
economically viable (to
both sides) are not
legally viable. If the resulting unemployed people are compensated by social insurance then the deadweight loss is transferred to higher income sectors of the population.
This is well known and understood by those knowledgable in welfare economics although a free-market proponent usually likes to point out that it creates economic "waste" in the system, which it does. The rationale for the creation of this waste is typically a prevailing social consensus that the free choice of the putative below-minimum-wage employees is impaired by inequitably differential bargaining power (IE they have none; the employer holds all the cards)
If minimum wage laws are there to express normative social consensus views then you might expect them to be proposed and voted in by concerned citizens without any help from special interest groups like trade unions. However this is problematic and it is a central prediction of public polocy and group theory that large diverse groups like electorates do not effectively organise themselves to further their own (group) interests. In these circumstances, a smaller highly organised group like a union can counterbalance this, so that society ends up getting what it wants.
None of this is pre-destined to work out for the best of course.