Siri remote not pairing. How to bypass Apple TV setup?

Just updating this in case anyone else has the same issue.

I was able to isolate the issue to faulty bluetooth on the Apple TV itself, through visiting a friend who had their own Apple TV 4th gen and testing that my remote could pair with their Apple TV successfully, but theirs couldn't with mine. As it happens I was also able to pair my remote with my Macbook Pro (hold Menu & + on the Siri remote while in Bluetooth pairing under System Preferences on your Mac) and control my Mac's volume, and play/ pause in iTunes.

I visited the Genius Bar and they were happy to replace my Apple TV with another 4th gen unit for free, despite mine being 1 year + 3 months old (apparently they only have a 1 year warranty). Thanks Apple!

As for my question:

Is it possible to somehow bypass this stage using the Apple TV remote app on my iPhone?

I found that yes, it is possible using the Apple TV remote app on my iPhone, as long as a) you had already removed all Apple TV pairings from the Apple TV remote app leaving the Devices list empty, and b) the Apple TV was connected via Ethernet to the same network as the Wifi your iPhone is connected to. In that scenario you need to then search for and "pair" the Apple TV afresh, after which you can use the Apple TV remote app instead of the Siri remote.

Hope this helps someone!

Thanks, but the Siri remote has been charged. I'm still unable to pair.

The CiderTV Mac app mentioned via the link you provided seems only able to pair with an Apple TV 4th gen via its Settings > Bluetooth menu... which I can't get to unless I have first paired a Siri remote at the Apple TV setup screen. There's no other way to pair an Apple TV 4th gen without access to Settings > Bluetooth unfortunately 😟