First, I don't think wanting to move to Italy or Switzerland, especially because it's for your children's benefit, is a stupid thing. I wish my parents had broadened my horizons like that when I was younger.
When I made the move to working remote, I was in a position managing a team of about 6-8 developers. But I was also still spending about 25% of my time coding. To go remote, my manager thought it would be best if I cut the managerial responsibilities but still have "unofficial" leadership responsibilities as a Tech Lead.
Fast forward about 9 months or so and I would say my responsibilities are largely the same as when I was managing a team. I essentially still am managing a team (although not in title). And my manager is definitely comfortable with my moving into pure management remotely if I want to.
This is all to say that you might be surprised with how much of the leadership role you're still able to do remotely.
I would absolutely first have the conversation with your current company about seeing if you can keep your current role and just do it remotely. Not sure how long you want to be in Europe but you might consider timeboxing the remote thing so that your company - if they have any reservations - don't feel like you're making a permanent change.
Also, if you do have to go back to a more individual contributor role, you can probably still maintain some leadership responsibilities, especially if it's with your current company.
If you have to look at jobs elsewhere, focusing on finding something remote will certainly give you more freedom and I'm not sure how easy it is to apply for (on-site) jobs overseas as I've never done that before. Remote companies are usually smaller and therefore their employees wear more hats, so someone with more senior leadership level experience applying for more hands-on roles will probably (my best guess at least) not raise as many eyebrows.