Italian here, while we are certainly not a progressive country on social issues, I do believe that the election result wasn't due to overwhelming popular support to the Fratelli d'Italia party and their views specifically, as much as to a general distaste for any other party and for the fact that we had to vote in the first place.
Only 63% of Italians voted in this election. It's the lowest attendance rate in our history. So that already indicates that a lot of people were not interested at all in who's governing, they probably were just fine with Draghi and wanted him to continue, and they just couldn't be bothered with the whole thing. The people who didn't vote are more likely to be the moderates, who feel neutral about a lot of issues, who probably would support social issues like gay marriage if it came to voting about it but aren't really passionate about it otherwise, who supported Draghi because he's a respectable figure with moderate views who seemed well equipped to handle the crises we are facing.
That 63% includes all the people who voted blank, and I suspect it was a lot of them. That's because many many people feel that the former parties in power (PD, Lega, FI, Movimento 5 stelle) were completely inept and useless when they were governing, did nothing for those who supported them, pretty much let us down in a crisis, and essentially sold off the whole country by causing the government to fall at such a critical point in time, just for a chance to be more involved in the next government when Meloni inevitably won the elections.
All of the parties that were previously in power underperformed at the elections.
PD did better than the others because they are the only "realistic" leftist party that we have. But even then I think that they have really detached themselves from their traditional base and that a lot of people who would have normally voted for them either voted for Meloni or didn't vote at all, because they don't feel represented by them anymore.
Fratelli d'Italia has been an opposition party for a long time and it was obvious from the start that they would win, if only because a lot of people are fed up with all the other major parties that have been in power for about a decade. Social issues like gay marriage or immigration were not at the forefront of the political debate in the last few months, which was mostly dominated by our stance towards the EU and NATO and in the Ukrainian crisis and the energetic crisis which are more topical issues.
TL;DR: I think that Meloni's victory says more about a lack of better candidates, a fatigue with the current status quo, and a general disaffection with the elections among the moderates that ultimately only gave more power to the extremists, rather than about the popular opinion on social issues.