Custody and more parenting time automatically means more child support, right?

Not in the slightest. At least not in my state. Child support is calculated based on a rote formula. If a guy gets more than "local rule" parenting time, then child support can go down, but giving the mother additional parenting time does not increase his child support obligation - it's only a one-way deviation. Technically the law allows it to work two-ways but there is absolutely no precedent in any jurisdiction I've ever researched in my state for an increase.

Many cynics believe women want custody and maximum parenting time precisely because that maximizes the child support payments to them

They're not cynics, this is real. I just finished representing a woman on this exact issue. Granted, the guy more than deserved it (constant acts of physical violence enacted against the mother and the child both; lying about his income; concealing assets; back-alley stock transactions with a privately owned corporations; etc.).

which they can spend however they want, not necessarily on children

So, this accusation always strikes me as potentially truthful, but most often ignorant. In my state, the child support numbers aren't based on future out-of-pocket expenditures alone. It includes things like rent to afford a place big enough for the kids; additional utilities the kids use; additional groceries to keep them fed; etc. In other words, basic living expenses that the mother would have to pay anyway are higher because she has the kids than what she would require if she were alone. The court builds these into the calculations. So, if you've got a child support obligation of $750/mo and the mother has to pay an extra $250/mo to have a 2-bedroom apartment instead of a 1-bedroom (or comparable situation for a house), then 1/3 of your child support obligation is built into this price differential. Tack in an additional, say, $100 for increased utility consumption and $300 for food, that leaves her with $100 left for clothes, extracurricular activities, school expenses, shoes, entertainment, etc. That's obviously not enough - and the courts expect mom to pay the rest out of her own income. Obviously I'm using round/estimated numbers, but that's how it works. Most guys mistakenly think that child support is just about the out-of-pocket stuff that you decide to buy and they forget about the cost differential for all the base necessities of just living.

How could a man "leverage" the female desire for parenting time such that she'd agree to "she gets more parenting time but in exchange he pays her less money"?

"I could fight for shared parenting with 50/50 time, but if you waive child support I'll let you have custody, I just want my time" and if she doesn't bite, "What if I let you have an extra overnight per week, as long as I get the after-school time that night?" Women eat this up. You offer that, she'll usually give you anything else you want financially - including a larger chunk of the retirement. The maternal instinct to feel in control of the children is insanely powerful. The guy can go off and control things in the world all he wants. A wounded woman doesn't experience that control the same way, so her children are her only source of power and stability in life - she'll give anything to keep that security.

About 80% of men I go against don't actually care about custody when they realize that custody has nothing to do with time with the children. They didn't care who picks the pediatrician or the dentist when they were married, and they don't care much more after the marriage is over. Really, those are the only big decisions custody affects. In some occasions it can affect extracurricular activities too, but those usually get spelled out as being joint decisions even in sole custody plans. "School decisions" gets talked about a lot, but in reality there aren't many to be made - you go to school in the district where you live and until high school you don't have much choice on classes, and even then the kids themselves are usually the ones who pick those with a guidance counselor, not the parents.

So, a guy has to ask himself at some point: is it worth $750/mo to be able to choose which pediatrician and dentist your child sees? She could, of course, pick an insanely expensive one and you'll always get stuck with 1/2 the med bills (sometimes more, but rarely) - but this hurts her wallet too, and a cheap doctor v. expensive one won't usually be a cost differential of $750/mo - so you're still saving.