…..this logic does not hold, or is at least more than compensated by an effect going the other way: legalizing the prostitution market increases its size so much that the supply of legal prostitutes is not sufficient and illegally trafficked ones are needed.
But there\’s absolutely a logic to it being possible. For prostitution itself exists because the supply of women willing to have instant sex with random strangers* is rather smaller than the number of men that desire that. If you relax some of the restrictions on demand from men by making it legal then yes, perhaps you are going to have supply problems.
*You can use the other justification for prostitution, the number of women willing to leave immediately after having sex is smaller than men would desire. But the shortage part still exists.