I’m gonna be the “Well, actually…” guy here and note that Battlefield 4 had private servers on consoles. It was partially due to them wanting to hit PS4/Xbox One launch, but they’re still in use today!
What DICE seemed to find from that though was that private servers and consoles aren’t the best mix since they’ve had DICE servers be the main base on consoles with BF1 and BFV. They do offer Community Servers and just recently added them to BFV although they’re not persistent. It’s only up as long as you host it. Still, it can be a thing on consoles.