When in summer? And for how long? Italian schools break up at the beginning of June, but most people have normal leave allowances so family holidays are more likely to be in August. June will not be heaving, although in Liguria there are a lot of retirees and some may have grandchildren in tow.
I agree avoid central Italy, not least because it’s a 4-5hr drive from Milan. I live near Lakes Como and Maggiore, also Orta. Depending on what you want to there are also family friendly places in all. When my DD was small we used to go to Colico, on lago di Lecco (top end of Como lake), small resort with plenty of waters sports plus a short drive to the mountains.
if you want to visit Verona etc then Garda will probably be your best bet. If you want mountains, the whole of Val d’Aosta is set up with summer trails, as are the Dolomites. I would pick one area, either Liguria, Piemonte, Lombardia, or else Veneto/Trentuno Alto Adige. I wouldn’t spend time driving from west to east. Unless of course you’re coming for longer than 2 weeks.
i have been to Courmayeur but in the winter. In the summer we visit the smaller resorts around Monte Rosa, which are a bit cheaper. Some have fairly small child friendly routes, although I’ve not tried them, we didn’t go hiking until DD was around 7, so she could manage the main trails.