Hi, since you already tried a replacement Siri remote (2021 model) this looks like you either need to get a new well shielded HDMI cable or there is something wrong with your Apple TV and you need to get it replaced.
When a poorly shielded HDMI cable moves 4K data at 60Hz, it can actually act like an antenna and broadcast electromagnetic interference, that interferes with Wi-fi and Bluetooth. The new Siri remote uses Bluetooth 5 the old Siri remote uses Bluetooth 4.
So while your HDMI cable might be doing just fine from a bandwidth standpoint (pixels aren't a problem), it's leaking EMF and causing the Bluetooth-based remote to have problems.
The fix is to replace your cable with a well shielded HDMI cable, try the cable Apple recommends.
New Apple Siri Remote 2021 Owner’s Thread + FAQs + User Guides + News.