I've been eying the phone sockets too wondering what I can do with them. Think i'll end up running fiber though because internet is 1.6 so gigabit would be a bottleneck.
As a side note - it's quite difficult to find white fiber cables. They're all bright colour so that nobody cuts them accidentally but I don't want a pink line running along the walls haha
No need for fiber for 1.6 gigabit, Cat5e can do 10GbE over short distances, 2.5GbE over longer distances.
Here: https://www.amazon.co.uk/Elfcam®-Fibre-Optic-Multimode-Duple...
It is flat and thin enough you can stick it on top of skirting boards/etc with tiny dabs of hot glue.