How to set up an iPhone or iPad to act as an Apple TV (4K) remote?
From the link you referenced, it tells you exactly what's required to use your iOS device as an AppleTV Remote


  1. Check that your iPhone, iPad, or iPod touch is on the same Wi-Fi network as your Apple TV or AirPlay 2-compatible smart TV


You need to have both the iOS device and the AppleTV on the same network.  


  Is there any way to make it work? Such as moving the iPhone or iPad 2 feet away from my Apple TV and have them "hooked up"? I guess one possibility was if that Apple TV sent a token to iCloud for my account long before, and can let me download that token to my remote app to authenticate it is the same user trying to use the remote functionality but I guess such functionality is not there (yet)? Is there any method at all?


Proximity and tokens are not methods that have been implemented so while it an interesting concept, it simply doesn't exist.


  A side note is, if a new remote needs to be purchased, I supposed it can still hook up to my existing Apple TV (4K), or else there really is no point in selling a remote by itself?


Yes.  You can (re)pair a new Apple Remote with your existing AppleTV.  You can get more info from the Apple Support document If you lost the remote for your AppleTV.