Apologies in advance if this seems a daft question
Can I use my BT Home Hub 3 as a "pure" router by conneting the red infinity socket to an ethernet port of another router?
Why?!, I want to use my trusted old router for the actual ADSL connection as it allows full remote access to monitor line performance/stats etc when I'm away.
No, that connection is on the WAN side of the internal firewall of the HH3.
Why not simply connect your old ADSL router to your broadband line, you do not have to use the HH3?
You can still use other BT wifi hotspots with a non-BT router yourself. Lots of people use Netgear routers, and are still opted in to BT wifi.