What can Vodafone see on my LAN?

[cleckhuddersfax]

So I've had Vodafone Pro broadband for a while. Recently I had a missed call/voicemail from them telling me my WiFi booster wasn't connected and they were contacting customers proactively. On the voicemail they called out some of my specific wifi devices with poor signal.

So, what exactly do they have access to see? Are they snmpwalking my router, or do they have admin access to the router and can access what's on my LAN? To be honest I'm quite concerned they can do anything past the WAN side of my router and it's making me consider swapping the router out for a separate device!

[HappyNomad]

I vaguely remember a bit of discontentment amongst users back when this change to their privacy was announced… no doubt still in these pages from the depths of time.  
At the time, I'm sure the agreed solution to the 'unwarranted and unwanted' snooping into our LANs was to simply bin the ISP router in favour of a third party device.  

I have been using a Draytek modem with a non VF router for a few years now and had forgotten all about this issue.