Clearly this can't be about people actually seeing the royals - very few visitors, for instance, will see Prince William on the(mostly closed to the public) two days of work he manages to squeeze into a week. If there is any draw, it's about the pageantry and the royal palaces. Well, guess what? We can still change the guard and all that stuff - in fact if we really wanted to save money, we could even do it with cheap unemployed actors, rather than wasting our military's time. And all of the royal palaces could be open all year round, rather than bits of them at times the royals fancy it. Oh, and there'd be a lot more public access in the Duchy of Cornwall.
As an argument, that's a busted flush. But I confess I've struggled in the past with the 'Yes, but if you have a president you'll just end up with another bloody politician as head of state,' argument. And what's going on in the USA, entertaining though it may be, is no encouragement. But I realised today what the model should be - university chancellors. They are the equivalent of a formal head of state, leaving the actual running of the university to the professional vice chancellor - and with a few oddities, the chancellors are excellent at their jobs. So let's model the British president on a university chancellor.
I don't know, to be honest, how they end up with such good choices, but here's one suggestion for how to do it. You would be barred from office if you had ever been a politician or civil servant. Each major party could put forward one candidate, plus a single public candidate, where anyone could put themselves forward for an online vote.
As always, there's fine tuning - come on, I thought of this on the train to Bristol. But you know it makes sense...