I'm going to forget people's names even though they remember mine
Do you have a hard time remembering people's names?
I don't care about the people anymore, I just don't want to waste my time with them
That is not very nice of you. Not returning courtesies to people is considered rude. People are just trying to be friendly so you should do the same with them.
It's not like their name means anything to me.
It doesn't mean anything to you but it means everything to them. Simple gesture like calling someone by their name makes a lot of difference.