Child support payments are usually calculated based on where the child sleeps in the evenings. While the welfare of the kids is very important, I think that hundreds or even thousands of dollars a month in support if the child lives full time with one parent is pretty much the opposite of “There’s no reason they can’t live in one house all the time.” When me and my wife were looking at separation, and she wanted full custody, I looked into this. (In Canada, fwiw) I could spend my every waking moment with the kids, pay for everything they need, and I still would have been liable for 60% of my paycheck in child support since they would be sleeping at their mothers house. (We reconciled, and we are doing much better these days now that my stress level is lower)
It’s just my opinion, but the only people I’ve ever seen pushing for sole custody are stay-at-home mothers who don’t want to have to go back into the workforce, and expect their ex to keep paying for it all. All in the name of “stability for the children.” Not to argue that kids don’t need routine and stability, but they can get that from two households too, without just one parent paying for it all.
In response to the OP, the best arrangement I saw when growing up was one set of parents where the Dad got all of them alternating weekends, and took each of them one night per week by themselves. The kids loved having one on one time with their dad, and it seemed to work out well.