Some of the OP's RJ11 runs apparently only have two wires connected, which may be a UK thing (or just a 'bad contractor thing'). Hence resorting to a solution that that can put broadband on two or four unshielded phone wires (probably based on ADSL-type signaling).
In the US, houses built to code in the last 30-40 years generally have 4-wire runs to RJ-11 jacks. TFA doesn't mention it but it is possible (although hacky) to gang up two four conductor phone wires into an 8 wire Ethernet run. It's a hack because those phone wires aren't even shielded as well as basic Cat5 but it can still work for relatively short runs. With more people using mobile phones, wireless phones or IP phones instead of POTS, that's leaving increasing amounts of in-wall RJ11 dormant. It might not work, but it's pretty easy to test and much easier than running new wires.
My latest house, built about eight years ago, came with CAT6A cabling for all of the phones. That made it really easy to just replace all of the RJ11 jacks with RJ45 (or 8P8C if you like to call them that). They've been providing POE+ gigabit Ethernet for the past seven years, and I'm about to upgrade them to 10GBase-T.
I envy you. Must be nice to open up a wall jack and be pleasantly surprised instead of bitterly disappointed :-).
I recently built a vacation property and could choose what to install. Initially, I was thinking CAT6A but after getting some good quality plenum-rated 6A to play with, I discovered it's surprisingly thick and doesn't bend very easily (minimum corner radii to avoid shield damage are actually specced) and it's not exactly easy to DIY new connectors on them correctly if you don't do it all the time.
So, after really considering the max throughput I'm ever likely to actually use, I decided to go with some good CAT6. It wasn't even a cost thing, as the difference was negligible. It was just practical reality. None of the runs are very long and I should be able to get up to 10gbps on 6A, although I doubt I'll ever use more than 2.5gbps (only using 1gbps today). One factor is the property is off-grid except for AC, so data is via StarLink anyway and fiber won't ever happen out there. While LEO satellite and terrestrial cellular speeds will increase in coming years, when more than 2.5gpbs is available - it'll almost certainly be priced to maximize B2B profit and we probably wouldn't pay that for a vacation house (because this is the US, not someplace that prices broadband rationally like Korea).
In the US, houses built to code in the last 30-40 years generally have 4-wire runs to RJ-11 jacks. TFA doesn't mention it but it is possible (although hacky) to gang up two four conductor phone wires into an 8 wire Ethernet run. It's a hack because those phone wires aren't even shielded as well as basic Cat5 but it can still work for relatively short runs. With more people using mobile phones, wireless phones or IP phones instead of POTS, that's leaving increasing amounts of in-wall RJ11 dormant. It might not work, but it's pretty easy to test and much easier than running new wires.