Not sure what you mean by “touch” in this context.

iPhone 13 with iOS 15 or later qualifies as “iPhone 4S with iOS 7 or later”.

You can use the software remote on iPhone with Apple TV (3rd generation), A1427 or A1469.

Set up the Apple TV Remote on your iPhone or iPad - Apple Support

Unlike later models, these legacy models will need pairing for this use by typing in a code from iPhone into the Apple TV box, using the physical remote. So, iPhone is not a 100% viable replacement.

You can stream using AirPlay content that is compatible with this model Apple TV box. However, some content providers require a later model Apple TV box for AirPlay.

Use AirPlay to stream video or mirror the screen of your iPhone or iPad - Apple Support