Apple TV (Gen 1)
It is more likely that the model is 2nd or 3rd generation. Apple TV (1st generation) is from 2007, and has no real functionality in 2024 (or any time after 2018-05) anymore.
Identify your Apple TV model - Apple Support
Find your Apple TV model number - Apple Support
The software remote in iPhone Control Center is compatible with Apple TV (3rd generation or later), not 2nd generation or earlier. Set up the Apple TV Remote on your iPhone or iPad - Apple Support
I have lost the physical remote.
You may buy an old style Apple Remote (aluminum) A1294 from select resellers that still have a few. E.g. on eBay or Amazon. Replacement remotes will have to be linked to the box, as described in: If your Apple TV remote isn’t working - Apple Support - Wayback Machine (second half of the page). Do both steps: the unlinking and then the linking. Menu + Left for unlinking, then Menu + Right for linking.
When I try to use the Apple TV remote on my iPhone, it gives me a 4 digit code to put into the Apple TV. Since I can't control the Apple TV I can't input the code.
The Apple TV (3rd generation) needs a functional remote to set up the Apple TV Remote widget in Control Center.
For the Apple TV (3rd generation), your options are:
Is there some way to reverse that process. Such that I input the code on my iPhone?
Unfortunately, the reversing of code display and code entry wasn’t thought of until the 4th generation.
Or maybe I need to reset the Apple TV?
That would make it a bit worse, as you’d have to configure the network settings once more, which needs a functional remote.