This debate has provoked a lot of interest, at least in me.
I live in Italy ATM and in my naive way am quite an admirer of 84 year old Sergio Materella the Italian president since 2015, he seems unpretentious, kind, inclusive and to give his office the dignity that it requires. In my uneducated outsider's opinion he appears to do a lot of good work (ribbon cutting, giving awards, visiting disaster sites etc). His daughter accompanies him on state occasions as his wife is dead which I feel is another nice touch. The constitution requires that the president is above all a force for national unity - given Italy's history and north / south divide that seems like a good idea. He doesn't get involved in the sometimes lively political debates that take place from what I can tell.
Provoked by this debate, I had a look at what happens here. It seems that elected representatives from all tiers of Italian government (Regions, Representatives, Senate) elect the President for a 7 year term - it is not a public election. Materella is now on his second term so has been popular, at least with politicians. The few Italians I have broached the subject with seem a bit less enthusiastic about him than me. His past was as a politician mainly with the centre-left Christian Democrats who governed Italy in the post war period before Belusconi came along and he held many senior positions. To some Italians I would guess that he probably looks like a career politician (from a notoriously corrupt party
https://muse.jhu.edu/article/36481) who is maintaining his proximity to the trough.
It would be the equivalent of say John Major, David Steel or Michael Heseltine being president of the UK. Despite its shortcomings, I would definitely prefer this arrangement to the monarchy along with a written UK constitution. There is no forelock tugging, no bowing, no pageantry, no mention of god and no presumption that he is anything other than an ordinary person. He represents the state with a sort of quiet dignity.