So...
Yeah, I don't really have that many close ties to my family. I mean, they do their duties like feed me and give education and stuff, but...
I can't say I am as close as I think family members should be. To be honest, I don't really care, but for the sake of SOCIETY, I'll try. ish.
For one, we don't really eat together. When I was young, I'd usually eat with my sister (I found my relationship with her a bit...conflicting). But my mom always kinda ate too late or too early, and my dad always eats late as he arrives home late from work. So, I kinda grew up my childhood more lonely (I didn't feel that lonely with TV) than others my age.
Apparently, big birthday parties where people invite everyone in the neighborhood and at school was a big part of American culture, but we don't really do such. For birthdays, we buy a cake, say happy birthday, and then that's it. Other times, we might just forget. One time, my sister forgot my birthday. Another time, I forgot hers. Another time, I remember when our whole family forgot my mother's birthday. She didn't really care and only spoke out in the end, but I mean...
Anyways, back to the eating thing. Around a month ago, my mom thought we should all eat together. However, after experiencing dining with my relatives during vacation, I find this a total nightmare. I mean, they bring up the same topics in the conversation like "eat more" or "study harder" or "learn more stuff blah blah blah."
Surprise surprise, that was the conversations that happened at the dining table. Especially targeted towards me, as always. I wouldn't have felt that annoyed, but 1. I'm an introvert, so conversations like these aren't really my cup of tea, 2. The conversations got old, and 3. We are all a one, big, short-tempered family. Like, no joke. Misunderstandings in my household (especially involving me) can result in 30 minute to an hour and a half arguments.

I would like to at least try to understand the other members of my family without getting into arguments, but my parents are generally stubborn and try to set my thoughts and ideals as theirs. (This backfires, obviously) In addition, I also have relatively low EQ. Like, I only know when they are angry (lol?). I try to listen to them, but then I procrastinate, and my mom's constant nagging always gives me an impulse to NOT do what she says. And whenever she says the opposite, it probably is in a mocking tone, so I get more pissed off. On peaceful days, I don't really care about it much (probably my mom too), and I just continue on with my life.

In general, is it really worth it to get to know them on a deeper level? I mean, I feel like it is fine, but I guess society says otherwise. I've read so many Chinese xianxia novels, and I can guarantee that they will talk about filial piety like every other chapter. (No offense to all netizens from the people's republic).

P.S. Those novels are really good though. The "You court death" scenarios are always fun to read.

I know what you're going through. Me and my mom almost never talk, And I can't trust her with anything. My dad is always calling me hurtful things,and my brother doesn't get pointed out as much. Everyday's The Same For Me. We never eat at a table, We bearly have proper breakfasts, Or any proper meals for that matter. Just know you're not alone!

If what you looking for is to bring your family closer together, I would recommend spending some time together. As you said, you all don't seem to like eating together and you guys always end up talking about the same things that aren't particularly enjoyable, but there are other things to do besides eating together. For example you can go out to see a movie with your family. That way you can all talk about the movie after it's done instead of talking about eating more/studying harder. On a side note too, I know that hearing your parents nagging you to eat more and study harder is super goddamn annoying, but they are ultimately trying to look out for you.

Also, do try your best to remember birthdays and remind your other family members of upcoming birthdays so they don't forget either. You say that your mom didn't really care when you all forgot her birthday, but I find that hard to believe. She at least cared enough to point out that you all forgot, and though she may not act upset, she very likely was at least a little sad that all of you forgot. From personal experience, my dad forgot my birthday once, and it wasn't like I was mad at him or got depressed because he forgot, but it was rather disappointing. Even though I never said anything to my dad about him forgetting my birthday, I was still a little upset. So I think the same goes for your mom about forgetting her birthday. She may not act like she is hurt, but she probably is.

Another thing, I would recommend you try to do is little nice things for them from time to time. Obviously, I'm not really familiar with how your family works so you'll have to see for yourself, but if your parents come home late, maybe try helping to do some chores around the house, like clean the house or vacuum around. Or maybe you could try cooking dinner for them sometime or buying them little gifts like chocolate or something from time to time. I know it might seem weird to randomly buy gifts for your mom or dad or sister for no reason or just one day decide to cook dinner for them, but these little acts here and there really help strengthen relationships.

At the end of the day, it's up to you to decide if you wanna take any of my advice. After all, you are much familiar with your family situation than me and it's up to you if you even want to put in the effort to bring your family closer since you say you don't really care anyways. But if you do end up trying to tighten your family relationship, I wish you the best of luck.