Just for info, I ended up with an Asus RT-AC68U and BT Openreach modem. Both came from ebay.
I plugged the modem into the wall, waited for the green lights, then connected the Asus.
About two minutes of very simple setup work later, I was up and running.
No dropouts, no issues - just rock solid broadband for the last couple of weeks.
The Asus has a load of tricks up its sleeve. One day I'll investigate some of those, but in the meantime I'm just enjoying having reliable web access.
Best £100 I've spent in quite a while.
Thanks for the advice.