The "do we want...." argument is actually an argument against democracy so I don't know why people make it. Essentially what you are saying is that ordinary people are so stupid that they should not be trusted to elect their head of state. Instead, we should leave it to luck who we get. Incidentally, we have no mechanism to get rid of them if they're bad, at which juncture I might point out that for nearly a quarter of a century Prince Andrew was second in line to the throne.
Obviously, I disagree with this anti-democratic argument because I would trust people to elect a decent president (or at the very least, vote them out of office at the first opportunity if they do a bad job). Bear in mind Trump only lasted one term in the USA and in the UK Boris Johnson only lasted about two and a half years before his position was untenable. Democracy works and when office holders misbehave or screw up they have to resign (e.g. Richard Nixon or Eliot Spitzer in the USA or Johnson or Truss here).
Also, it depends on what kind of presidency you have and how they are elected. If you have a President who is actively involved in the day to day management of the executive like in the USA or France, then you will probably have a Presidential Republic and get a politician with a track record of working in government. However, given the UK's constitution, it is much more likely you would have a Parliamentary Republic where the President is more of a figurehead and open to people from other walks of life. Boris Johnson could be a risk if he was standing against Tony Blair, but I wouldn't fancy his chances if he was up against someone like Stephen Fry for example. Similarly, it would be wise to have run off elections so only someone capable of achieving more than 50% of the vote in the final ballot could win, which would safely rule out people like Trump who relied on an electoral college or Boris Johnson, a man the majority of the country couldn't stand from the day he became PM.
There's one other very important point too though. People complain about a President Blair or President Johnson but here is the rub. Rishi Sunak already has more power here than Joe Biden has in the USA or Emmanuel Macron has in France because of the Royal Prerogative. There's a constitutional convention that he should not do so, but he can take the UK to war without anyone else's consent. He can award honours to whoever he likes; some of the emergency responses to COVID-19 such as the VIP fast lane for Tory donors to supply PPE were the result of these powers too. So actually one very good reason to do away with the monarchy is to take powers away from the Prime Minister and hand them to Parliament.