My monthly plan payment would not go down if I brought my own phone.
Your particular plan might not but there are BYOD plans that are significantly cheaper than anything you'll get on contract
There are plans available where that is the case. So the solution would be to switch plans or carriers to one that doesn't bundle the phone price in.
HN apparently can't fathom that people pay for Verizon because the service is good. I get Verizon free through work and sorry it's noticeably faster than Visible. The discount plans aren't actually the same but cheaper. If you were going to be a Verizon customer anyway then the free phone is actually free. You don't get a discount for BYOD and the service is the same price whether you take the phone or not.
You sure about that? Look up Visible, Mint Mobile, Total Wireless, US Mobile, Tello... Same carrier networks, same quality of service. You can even pay a bit extra for prioritized data and other fancy features. You can get basic unlimited plans for $15-30 and premium plans in the $30-50 range vs $100+ at the big carriers. The only difference is that you aren't paying for your "free" phone.