Actually, cheating and keeping other people on the side happens all over the world in every culture, not just in the West. Are we not dealing with the standard definition of marriage and common-law, meaning two people co-habitating and possessing official status of their commitment through documents like legal and religious marriage certificates or living together long enough to fall under legal frameworks (e.g. common-law)? As far as I know, having a mistress is not common-law because the man does not live with her every day and therefore would not legally fall under the framework for common-law.
While Islam allowed divorce, the Christian church was more resistant - precisely for the reason that they wanted to keep families together and have the couple work things out. They saw this as being in the best interest of society to avoid too many cases of single-parent families etc. It has turned out that many single-parent families exist today, but that is life. Women having access to good education and being allowed to participate in the workforce means that they can support themselves and any children they are left to raise alone. That is certainly a good thing. In polygamous relationships I find it hard to believe that a man can truly be there to support and raise so many children. He'd be spending so much time making enough money to financially support everyone. In essence, the women raising the children are pretty much doing it themselves anyway... sounds like single-parenting to me.
In cultures that accept women marrying more than one husband, the paternity of offspring shouldn't matter because there are likely laws e.g. about property ownership etc. that wouldn't rely on who the father is. Women are supposed to accept the other children from multiple wives and in some cultures they are raising the children together, the same should be expected from the men as well. In modern times genetic testing can be used to determine paternity, if it is deemed important.
In old times a shortage of men would have been a problem since men were the breadwinners, but nowadays women don't need to marry because they can work to support themselves, and they can find husbands elsewhere. Simply giving women more choices than marriage and producing children reduces the need for them to consider marriage with a man who already has some wives. I don't really see any other advantages to polygamy. One man and one woman balance out - too many men or too many women is not balanced and makes for lots of problems, in my opinion. Indeed, people who take on more than one husband or wife have guts!