his daughter's "sacrifice" of not wanting to get married seemed to be more selfish because she seemed to have her own reasons for not wanting to get married
She is a good daughter brought up with traditional ideas in mind, e.g. parents are more important than anything else. Her deep love for her father is partly the result of this upbringing. When a Japanese woman got married back then, she would leave her family and become a part of her husband's family. Obviously, this wouldn't mean she couldn't ever visit her parents, but it indeed meant that she would no longer live with them. She would essentially, and largely symbolically, change her family. And cutting these ties, again symbolically, is something Noriko is grappling with.
It feels like the movie is trying to convince us that she's willing to sacrifice her happiness for her father
A traditional Japanese woman is supposed to be unselfish and sacrifice her happiness for others. Period. If not for her father, then for her husband. However, once a girl reaches a certain age, she is expected to get married. There were both some men and women who never got married but that wasn't that common. Marriages of love were not unheard of, but arranged marriages were still going strong, and sometimes people who reached a certain "late" age simply got married to the first available suitor if only to escape the pressure of society. And then sometimes they were only married on paper and both had romances they both accepted... But then some never married period, resisting the pressure or being lucky enough not to be forced to. What's interesting, Setsuko Hara, the actress who played Noriko, never got married, so you could try and analyze this in a meta way...
And even after she was told that her father is getting married, she was still pushing for things to stay the same, so she can stay with him and take care of him.
Because that was Noriko's biggest wish even though she wasn't particularly opposed to the marriage. She not wanting to change would imply being traditional, but at the same time, she dared to confront tradition by being reluctant to marry... Many girls were reluctant, unsure, afraid, etc. but in the end were pressured to marry anyway, fulfilling their duty to society. But times were changing and more and more women started being independent and 'modern'. But that's another topic. Noriko not wanting to leave her father does not necessarily mean she doesn't want to marry. It's just the circumstances and the societal & traditional mantle that goes with the marriage that she is scared of.
everyone seemed to be pushing her a little too hard to get married, especially her aunt. I don't know why they didn't let her stay and take care of her father if that's what she wanted to do.
Because that was expected of her. Period. In such traditional societies, you just got to marry and have children. Your choice not to, if any, was dependent on many things and could have many results. Family repeating 'you're old, you must marry' is not that terrible. This doesn't apply to this film, but for poorer families, it was of utmost priority that daughters get married as fast as possible so that the family has fewer mouths to feed, etc., etc. Japanese society back then was traditional, but some very progressive voices were there, too, so it wasn't as one-sided as you might think from my post. Read about the practice of Sati in India - when a husband died, a widow had to burn together with her deceased husband's corpse. If she didn't, she was most probably expecting to be mocked, abused, and thought a terrible, undeserving wife. But accounts wary... It's just to point out the Japanese weren't that bad
But still, Ozu is a traditional director, so the conflicts in his films are traditional, too, even if he is extremely aware and up-to-date on the trends.
But at least in the end, it seemed like everyone was happy.
Both happy and sad. And this bittersweet feeling is what makes this (and other Ozu films) so powerful. Mono no aware. The father is happy because he did his job: he brought up his daughter to be a respected woman and wife. The father is also sad because he essentially lost his daughter and won't be seeing as much of her as before, so he is lonely. And him peeling an apple in an empty house is a powerful display of that loneliness.
Noriko is also both happy and sad because of that conflict of being a good daughter who provides and cares for her father vs. being a good wife and following what's expected from her.
On a minor note, I wanted to see the guy who "looks like Gary Cooper", but we never got to see him.
Every girl who's seen the movie points that out. There is a mention of an American boyfriend in one of later Ozu films, too, but we never see him either. Let your imagination do the work.