I'm not sure why the OP assumes that UK citizens living abroad don't pay UK tax ?
Firstly , everything that I do is in accordance with the laws of both the UK and France.
I am a lifelong and intend to continue to be a UK citizen.
I too am British living in France. I moved when I was 67 years old , three years ago. We are not sure whether we will stay abroad or return to the UK but even should I die abroad I will still have spent nearly 80% of my life in the UK.
I am a lifelong and continuing UK taxpayer and for the 40 years that the UK was a member of the EU , I was, in my opinion, indirectly , also an EU taxpayer.
Can I repeat, very clearly, that I am a lifelong and continuing UK taxpayer and that prior to my retirement I worked for nearly 45 years, 95% of it full time.I have paid and continue to pay plenty of UK tax.
I also pay the French tax that is legally due to the French government.
I am entitled to retain an interest in my country and how it is governed and as it happens I have just been re-granted , this very day my Overseas Postal Vote so that I can vote in the forthcoming general election.
I have family members in the UK and I take an interest in the fate of the country also with a mind for their futures, especially the young ones. If we ever got someone like Trump in the UK then I would be delighted to have the opportunity to try and vote him out.
Other posters have pointed out that in the digital age one can remain fully informed and that many countries including the USA allow their citizens abroad to vote in US elections.
Among other examples I consider my right to vote just as valid as UK citizens living in the UK that don't bother to vote at all , particularly in General Elections.
I am not , post Brexit , allowed to be politically active in France.