iOS 12.2 Control Center Remote Control Finds no TVs
I can’t say for certain which of these mattered but here is what led to this bing fixed for me. 


Update to iOS 12.3
Update AppleTV to tvOS 12.3
Reset Network Settings on iPhone 
Add AppleTV to a room in Home.app
Delete all paired “extra” remotes on AppleTV (apps and other devices)
Give iPhone new, not-previously used name
Re-pair iPhone Remote app (which, remember had worked this whole time) to Apple TV


I suspect it was steps 5-7 that were all I needed, and I’m bummed that I nuked my Network Settings, but the Control Cemter widget works for me now.