Can you recommend any mountain areas for a larger family?

I think Chiang Rai would be a great choice for a large family, particularly if you're willing to splash out on tours or a private van for excursions outside the provincial capital city.

Chiang Rai is a bit further off the beaten path than Chiang Mai, but still relatively easy to reach and with plenty of local travel offices and tour operators that can assist English-speaking travellers.

The provincial capital is full of photogenic temples and museums and also has a great night market. There are loads of trips from the city, including the Tham Luang Cave that was the site of the incredible rescue of 12 boys and their football coach back in 2018. Other good options include mountain viewpoints like Doi Tung and historic Mekong River towns like Chiang Khan, or if going further up into the mountains, the tea-growing outpost of Mae Salong.

Do keep in mind that, as with virtually any mountainous area in Thailand, travelling involves winding roads with switchback turns. Also the entire north gets hit by bad air pollution late in the dry season. I do not suggest going up there in March or April and even in February the air can get bad as farmers start to burn off the stubble in their fields.

One other suggestion is Kanchanaburi, which isn't too far from Bangkok and can be fun for families especially if you're interested in the World War II history. If you go, the Hellfire Pass Memorial and Museum is not to be missed. Kanchanaburi province has some of Thailand's best waterfalls, such as Huai Mae Khamin and Erawan. While it might be too far out of the way for a large family, the lake mountain town of Sangkhlaburi is an enchanting place.