I'm from a very large family. I have 6 siblings, 11 nieces and nephews, one great-niece. (Also, My dad had three siblings who also had big families, so lots of cousins). We never give presents or cards on holidays or birthdays- it would rapidly become overwhelming. There is an understanding that people aren't expected to show up for every birthday party. I know approximate dates of birth and if I looked at my calendar, would know exact ones. This doesn't mean I don't love them or care about them.
Birthday party for little kids might be party with school friends (family attends if space/if desired). Younger kids get a few presents for birthdays/holidays, but certainly not a gift from each person. Other birthday parties are usually dinner/lunch/get-togethers; if people can't make it there's no guilt or anything. For grandparent (my parents) major birthdays eg. Dad's 80th, or weddings, there is a big get together and everyone tries to make it.
Presents are given randomly if you see something the person would like or if you know the person needs something. Major life events (eg. HS/College graduation, baby born, etc.) people may send a card and check - but not necessarily. Thanks to whatsapp we have a family chat where we share pictures, birthday wishes, congratulations, small and large life events (eg. dog learned new trick, vegetable garden planted, niece got her first job, other niece had a baby, nephew took a trip, etc.). However, it's not up to me to keep track of each child/ sibling and what they're doing. Either the person themselves reports, or the parents, or another person in the chat who recently spoke to them or saw them.
Also, by necessity in a big group, you are closer to some relatives than to others. For one brother, I know his kids super well and everything they like and dislike and their bedtime routine and I see them regularly. For another, I know names, month they were born, approximate age and I get pictures and stories from other relatives, but his kids wouldn't recognize me if I came to see them. I think this is probably normal in a big family. Again, it doesn't mean I don't love them or care about them. Even the ones I'm not close to I would absolutely take in if they needed me to. At one point, my sister sent her son to live with my brother. The son had probably only visited with his uncle every other year, but my brother was in a place in his life where he could take on a child. I guess my point is that not sending a card/present for every occasion and not attending every birthday party doesn't mean you're "awful" and you "don't care".