Ask the many Mormon male youths forced out of Fundamentalist Mormon societies where polygamy still is practiced (e.g., Colorado City, AZ); the older male hegemony wants a monopoly on the "upcoming crop" of coming-of-age young women. (My statement reflects that hegemony's commodification of women, not my own.)
You might object that marriage in itself is an outdated/outmoded, and we should all live polyamorous lives. Marriage has been a relatively stable social/legal contract for millenia, and I think attempts to make it irrelevant or obsolete will always fail.
What would that reasoning be? Who cares with how many other people I have what kind of relationship if everyone involved is okay with the situation? I never seriously thought about it but I think it would be a nonissue if I lived together with several women. Maybe the neighbors raising eyebrows and the state would not allow to officially marry several women but how cares? I would say for all practical purposes polygamy would be just fine.
I wouldn't and couldn't prevent them from talking to one of my women and convincing her that it would be a better choice to leave me and enter a relationship with them. And you could of course make the same argument for a monogamous relationship, you are still removing one potential partner for everyone else from the pool of potential partners - but you are of course not really doing this, at best temporarily. And what a single person thinks or feels never really matters anyway because a relationship is a mutual thing between at least two people.
How you prevent the situation where wealthy/powerful men begin to monopolize the pool of available women? What happens to the men that now have a harder time finding long term relationships?
To me, it seems it would accelerate the wealth gap we already see today.
Edit: Here is a good summary of some of the concerns I would personally see with a system. From the Supreme Court in British Columbia.
It is a interesting question whether you have a right to a relationship or if it is at least desirable to have a policy that makes it more likely that everyone can find one. I never thought about it.
My views regarding wealth distribution and the like would easily be associated with socialism and communism by many (Americans at least). But here my first reaction is more like sorry for you if you can not find a partner because of your social status. This worries me a bit because you could easily argue that a relationship and having children is in some sense more important for humans than being wealthy. I really have to think about it for some time.
The only thing I am already pretty sure is that there is nothing that justifies a right to have a relationship, i.e. there is no justification to force someone into a relationship with someone if for example this person is simply a really unpleasant person. But is it desirable to level the playing field as good as possible? I can't say ad hoc, but my intuition is no.
