For those worrying about a sinister motive behind the new rules, it's really not, it's just the typical bureaucratic 'cock up rather than conspiracy'.
The issue is:
A British citizen should ALWAYS have entered the UK on a British passport, but these rules were never enforced strictly at the border (and potentially still aren't - a British citizen cannot be refused entry to the UK);
Until recently it would never be picked up, because a non-visa national (which is 85 countries) just had to get on a plane with their passport, and no further visa paperwork would be needed, because their entry as a Visitor would be determined at the border. It would be an eagle-eyed immigration officer who may spot that someone is a British citizen, and then if they are, they are going to enter them into the UK anyway.
But now an ETA is required, which the airlines will demand to see is in place before they allow boarding - because otherwise, under UK law, they are liable for paying for the removal of the person if they are refused entry to the UK.
BUT - if a dual national applies for an ETA they will be asked questions about their family history and any other nationalities - and if the ETA questions flag up the person is a British citizen, the ETA is not issued - because under UK law it is prohibited to put a condition/restriction on the right of residence of a British citizen.
If the ETA is not issued, the airline is not going to allow a person to board the plane - and any airline official is not going to be an expert in British nationality law, which is incredibly complex (hell, I've been a UK immigration lawyer for 15 years and I always pass on nationality tracing enquiries because it's just too specialist).
So if a dual national rocks up at Calais/Eurostar with all of their paperwork showing that they are a British citizen, albeit without a passport, it is very likely they will be able to board their ferry/train because the British immigration officer granted them entry before they get on the transport.
BUT, as many on this thread have pointed out - so many people are British citizens without even realising they are, or have no interest in having it - but it's not as simple as saying 'no thanks' - you have to go through a proper procedure to renounce it. I've had to just stop my next door neighbour spending £1,200 on a 'registration of a British citizen' application for her granddaughter who was born and lives in the US - because I could tell her that her granddaughter was a British citizen by descent at birth and no registration is application is required. She assumed, because the child was born in the US, she was not a British citizen.
I don't know what the solution is for this. I think it's ridiculous that they are insisting on having a passport or certificate of right of abode (which costs £500) for someone coming on a holiday, but as a solicitor I can understand the legal basis for the request. I think the only thing they could do is make a mechanism to obtain a notation on the ETA which satisfies the airline but indicates to the immigration officer that the person is a British citizen and should be entered into the UK with no restrictions.. but then the Home Office answer to that would be 'well that's what the passport or right of abode certificate process does'.