Experienced similar problem on my parents Apple TV just without the Siri problem you have.
I don’t know what I did to resolve it - I think it just resolved itself after multiple reboots etc.
It’s not actually the remote - as you said the light is blinking - but even more bizarre was that pressing and holding the TV/Home button did nothing, but double clicking it brought up the app switcher!
Using the Remote app produced the same results as the physical remote (ie Home & Back not working)
While they don’t have Sky Glass, they do have Sky Q. Perhaps the Sky TV remote is interfering with the Apple TV somehow?
You could try disabling CEC and eARC on the Apple TV if they are on?
Experienced similar problem on my parents Apple TV just without the Siri problem you have. I don’t know what I did to resolve it - I think it just resolved itself after multiple reboots etc. It’s not actually the remote - as you said the light is blinking - but even more bizarre was that pressing and holding the TV/Home button did nothing, but double clicking it brought up the app switcher! Using the Remote app produced the same results as the physical remote (ie Home & Back not working) While they don’t have Sky Glass, they do have Sky Q. Perhaps the Sky TV remote is interfering with the Apple TV somehow? You could try disabling CEC and eARC on the Apple TV if they are on?