I don't think the bit in bold logically follows. Surely there are already countries inside the Schengen zone which don't require carrying ID or registering your whereabouts with the authorities? So from the Schengen point of view having the UK inside wouldn't present any new problem. The problem would be for Brits who could then travel into the rest of the zone without ID (given the lack or legal requirement for it here and the lack of internal Schengen automatic border checks) who'd then be caught out if in a country that could demand ID.
Well - ID cards are issued in Italy, but it's not compulsory to carry one around with you.
Everyone in Germany has to have a card, but doesn't have to carry it around.
Austria issues ID cards but having one is not compulsory, let along carrying it round to show on demand.
French citizens are not *required* to carry along identification documents and possessing a national identity card is not mandatory.
But in practice, it is difficult or even impossible to carry out many acts of life without a national identity card or a passport (such as taking an official exam or opening a bank account).
Some activities are getting more difficult to undertake in the UK without showing some photo ID (most commonly a passport of driving licence if you have them). However, given compulsory ID cards aren't "a thing" here, it's ultimately difficult for an organisation to insist on them. I have, for example, in the past been told to present photo ID to open a bank account and to stand for election to a public body; in both cases I refused, on the grounds that it cannot be assumed that people have such things, given it's not legally required. In both cases there was a lot of argy-bargy for a period of time (some weeks in one case) - but in both instances I was eventually allowed to go ahead without such ID.
I think that one of the reasons that an attempt to bring in compulsory ID cards in the UK, early this century, was unsuccessful was because of the tens of thousands of people who signed a pledge to non-co-operate with the system if it was introduced; significant open defiance would have made the plan unworkable. Indeed, after ID cards were introduced here during the Second World War, but not scrapped at the end of the war, they were subsequently abolished precisely because, with the perceived need for them having ended, people started refusing to show them. A prosecution of someone who refused was the catalyst for their abolition.
I guess many people simply see such things as ID cards as "rather un-British"....
