At a certain point, don’t you just have to say “We’re the parents, and what we say goes”?

Yes, but only if you’re saying what is in your kids’ best interests. If not, throwing your weight around is not an admirable legacy, especially as you’re already forcing enough upheaval down their throats. The more you give your kids a voice in good times, the less they’ll one-up you with the other parent in bad times. Write into your agreement how you will cooperate to make tough decisions so that you don’t undermine each other or squelch the kids’ legitimate voices.