Not 35+, think of mid/late 20s as it used to be about 20 years ago. Peak health, some disposable income, educated, but also good age to bear a child. If you start having children at 35-40, it’ll be harder for you to have 3+ children, but now for health reasons.
Up here in Canada, according to statistics, only 20% of couples want to have 3+ children. If I recall it correctly, it’s not that they have it, but plan to, which might not even happen due to socioeconomic reasons.
I don’t think majority of young people want to have 3 children though, as you mentioned. Even if I do an extremely unscientific look around in my circles, the number of couples who want that is 0. I obviously might be in a bubble, but it’s a fairly common sentiment especially for my age group.