They put it down to ‘Italian bureaucracy’ in that snippet, but countries are perfectly entitled to set the criteria for which non-citizens they want.
It's a bit more nuanced (and so much worse than that) though.
EU countries are not entitled to set conditions on citizens of other EU countries. That why when David "cunt" Cameron wanted to stop benefits for EU nationals in the UK, the only way to do it was to ensure the same benefits were not available to UK nationals. Luckily he had zero problem with shitting on UK citizens, because that is the Tory parties mission statement.
So to revise the story - Italy could not impose any conditions on UK citizens as members of the EU. However the moment we decided we wanted out of the club, those rules fell away and the UK because a third country. That is a country that is neither a member of the EU nor covered by a single overarching treaty. And it's that way because at every single possible turn, when the EU bent over backwards to engage with the UK, we told them quite unequivocally to go fuck themselves, their fathers were hamsters and their mothers smelled of elderberries. And still we went back to be taunted a second time.