I really don’t understand why it’s hard for Vegeta to tell his wife and children that he loves them. Is he that arrogant that he thinks he is tooo good to tell his family that he loves and cares? No shade
Both in a Watsonian and Doylian sense, it's cultural.
He's from a Japanese series, and iirc saying I Love You to your wife and kids isn't really a thing in Japan. You might notice that Goku's not riddled with pride, but he doesn't say it either (the closest he gets in the manga is saying that he "loves that part" of Chichi when Vegeta mentions Saiyans like strong women). Both in-universe and beyond the fourth wall, culturally they're more about actions than words.
From a strictly in-universe standpoint, while there are plenty of cultures like that here on Earth, Vegeta is not from here. He's not human, and he was raised in a military -- verbal affection is never going to be instinctive for him. He expresses his affection by trying to keep his loved ones safe (keeping Kid!Trunks out of serious battle, silently putting his body between Future!Trunks or Bulma and anything he perceives as a threat are great examples of how he publicly expresses his love for them.) Many cultures treat nonverbal affection as normal, and there are even some cultures that express love through comfortable silence, and I think that sharing space without needing to fill it is something more in Vegeta's wheelhouse.
Just wanting to be physically near his family when he could be literally anywhere else is an expression of love -- which is why I'll die mad that they changed this scene in the anime.
Pride certainly has something to do with it at the beginning, but imo it's less I'm Too Good For This and more This Is Not How Things Are Done (although depending on what era of Vegeta we're talking about it's probably a mix of both). Vegeta spends the whole first series unlearning cultural propaganda that ultimately did nothing but hold him back, and by Super('s manga) he's cooing over his new baby in front of God and Everybody.
tl;dr: Vegeta tells his family he loves them all the time, and they do the same for him, they just communicate it in less direct but more substantial ways. Plus, like I said in another post, talk is cheap when you're raised by liars. "I love you" wouldn't mean anything to him unless it was earned first anyway, and by then it's just an 'I know' for everyone involved.
Also like I said in another post, Bulma probably says it to everything around the house from the cat to her car keys all the time so he probably doesn't think it means all that much on Earth anyway, haha
