I think a key reason is that a lot of people who reach the point of taking their own life but don't follow through go on to recover and live a happier life. All of those people are so thankful that they never went through with it. Choosing life in that situation means you give yourself that chance to live a happier life - it won't be perfect, and not everyone does get there - but many do. Staying alive is not a permanent decision, and so you could give it some time before making a decision. But following through is permanent, and there's no going back.
Also, as much as we might want people to not be affected by what we do, the truth is that we can't control that. We live in an society of interconnected interactions and relations, so everything we do affects someone else in one way or another. You might not want your family to care about you, but they may do. The funny thing about family is that sometimes you can do your utmost to cut them off, but they'll never stop caring about you - unconditional love is a powerful and resilient thing.