I believe that it would be easier for someone to be happy if they do not feel anything towards the suffering of people they do not know.

Emotions aren't modular in reality. Your argument seems to boil down to "caring about suffering is a net emotional negative, so happiness is easier without it". That's a structurally sound argument but also an unrealistic idealization. In actual human psychology, it's very unlikely anyone could lack the empathy that compels most people to care (at least in the abstract) about the suffering of others while simultaneously possessing a full and unimpaired capacity for happiness. In fact, a lack of empathy correlates with conditions generally viewed as limiting emotional responses (e.g. autism and psychopathy) - the exact opposite of what would make happiness, or any emotion, easier to achieve.

Of course, it's not clear why this is. But the empirical evidence strongly suggests that empathy is an important correlate of general emotional health, and it's silly to throw that evidence out the window just because it's slightly counter-intuitive. The fact that empathy can directly cause sadness doesn't mean it isn't also indirectly critical to happiness - we have to good reason to believe that it is.