The new Siri Remote (3rd generation) is compatible with Apple TV (4th generation or later, including 4K), when running tvOS 15.4 or later. You can pair the remote using the button combination Back (Menu) and Volume Up.
If this model Apple TV box isn’t running tvOS 15.4 (March 14, 2022) yet, then you may need to borrow an older remote temporarily, to do the update first. The current version is v17.6.1 per August 19, 2024.
To use the software remote on iPhone (or iPad), you would first need to connect to the home network, which may require a functional remote. Unless you have ethernet (!) available.
If you have or can borrow an old style Apple Remote A1294, then that would be enough to do an update.
If you can’t update at all, then ask Apple Support if there are possibilities through Apple Store or AASP/AAR.
Siri Remote (3rd generation) technical specifications - Apple Support
Apple TV remote models and their compatibility - Apple Support