I am in America and the mother of five DCs. Many of my American friends are also parents of families which would be considered large by MN standards.
First of all, I'd like to state that I have always found American children to be more calm, less 'giddy', less whiny (for want of a better word) and more independent of their parents than British children. There are definitely exceptions to the rule, but in general there is an expectation that American children will fall in with what the group or family is doing. This goes for families of average religiosity, hard-core religiosity, and none at all, and it goes for American families of all sizes. I've definitely seen badly behaved American children, but in general they tend to respond well to parents, caregivers, camp staff..
The biggest family I know have eight kids with about 18 months between each baby. Average religiosity - the parents are both only children and wanted the opposite experience for their family life. Father owns his own business. Mother sahm former nurse. One SN DC.
The key to their family's happiness (and they are a happy, well adjusted family) is fantastic teamwork in the parents, a shared idea of discipline, of the values they hold and wish to have the children absorb. There was 1-1 time between both parents and each child every week. The parents each got time off for themselves each week too. They used 123 Magic with consistency and success.
In general, their parenting was all well planned - definitely an 'all in' style of parenting. Neither of the parents ever shirked. The father was as involved as the mother in childcare and managing logistics. Both were very gung ho about taking the kids places both together and alone. The kitchen featured a huge family calendar, a big noticeboard with kid photos and drawings. Both parents volunteered in their kids' sports and made the effort to provide extra curricular activities and do all the schlepping that required. They went on long road trips long before tablets were a thing, stayed in cheap hotels for holidays.
We did too. The DCs brought books and walkmans and we listened to baseball on the car radio and played games like state license plate bingo. I'm talking about entire days in the car here. Ohio is a bloody enormous state to drive through, ditto Nebraska. Kids were expected to get with the programme and entertain themselves to a large degree, obv with some input as necessary from parents.
Clothes were second hand and handed down. In my community friends tend to hand clothes and winter gear on to each other - I put bags together for many friends and they did the same for me. Elementary school uniforms were handed around.
Many Americans tend not to be fussy about brands, or clothes with the odd hole or paint stain. Kids for the most part pick out their own clothes and dress themselves for school and for the summer and winter breaks. There isn't the same level of micro management of children's appearance that you find elsewhere. The comment that the children on the plane were well turned out is not one an American under age 80 would consider relevant.
I think if you plan to have a larger family you are aware that you're taking on a lot of work, and that you're going to be outnumbered as soon as your third child is born. You learn to look ahead to natural consequences for yourself of every decision you make in parenting each child, and you multiply that by a few digits more than the number of children you have because consequences grow exponentially in larger families. You can't afford to do the easy thing in the heat of the moment with one child if it's going to mean the other 4-5-6 or more are also going to end up clamouring for some treat or special treatment. When you say no you have to stick to your guns. You can definitely get away with more when your family is smaller because the effects are easier to manage.
The result of consistency is that children see they're being treated equally, there's fairness, you don't see a lot of competition or jealousy or grudges, and playing up isn't going to get you anywhere. Kids see themselves as a little tribe too, and understand that the tribe does things a certain way. It's a little like the way kids get used to being part of a large group in school.