Good question!
I mean, in the Westerosi context, there’s still quite a bit of violence and other causes that deals with younger sons – I mean, technically there hasn’t been a fighting war in any of the Dunk & Egg stories, but for all their lighter tone, they have a pretty high body count – tourney deaths, plagues, “pissing contests” between local lords, bandits, stupid coup attempts, etc.
But in terms of how the social order would react, it’s a bit tricky because an extended period of peace probably also means an extended period of prosperity as well, if only because the opposite tends to intensify resource conflicts and thus lead to war. And prosperity is a great social lubricant.
When the harvests are good, trade is up, and people have cash on hand and good terms of credit, it’s easier for the social order to deal with surplus kids – get them dowers/dowries to smooth the way for a marriage that wouldn’t have made fiscal sense otherwise, give them jobs around the castle or pay a neighbor to take them off your hands or send them to court, or even set them up as a landed knight or cadet branch if you’re particularly rolling in it. And yes, I imagine you’d see quite an uptick on younger sons and daughters getting sent off to septries and motherhouses with generous donations, as well as an increase in acolytes and novices sent to the Citadel.