Regarding your second point, I'd like to point out your particular phrasing which makes clear your point of view, that you consider a multi-generational house "someone else's house". It would be your home that you are living in, even if it is not your name on the deed or rental agreement. Now, whether you can develop your own relationships and life and way of doing things has to be independent of whether you are living with family. For every person that is an example that living with family is a burden, there surely is another example where it is a benefit. The growth in your relationship with your family can be hindered or accelerated by your time with them, I don't think there is a one-size-fits-all answer to that.
In my experience what happens is that the "family" raises the children.
In intergenerational households - in Asian cultures at least - it is still generally the older generations calling the shots, in patriarchy/matriarchy sense. In practical terms it depends on how much financial power each generation holds that determines how much say they have. But in Asian cultures age do grant at least some implicit authority.
Given the above cultural context, tensions can arise if the way of living/child-raising are not aligned. I would say this resolves how normal conflicts get resolved. Those who have the most time and influence usually win out. Although culturally people understand they have to pick their battles for the benefit of daily harmony.
I think a way to think about how the living arrangement works that may be more relatable is to think about a multi-generational family business. If you watch shows like Chef's Table there are stories were children take over the family's business by more-or-less following in a prescribed footpath, while others conflict and find that they have to strike out on their own.
On a day to day basis though, it's just like living in a very close knit community - you could say a family.
I think this is the simplest answer to why most people are opposed to it - they fundamentally don’t want anyone else calling the shots on their lives to any extent.
How do you have a grown up, likely physical, relationship with your partner... when your parents are sharing the same rooms? Do you just hide in your bedroom the whole time?
How do you develop your own relationships and life and way of doing things and way of raising your children, inside someone else's house? I can't understand that but I'm open to hearing about how.