Look, you are a nice person. You know that, we know that. That's not the problem. However, just because you're a nice person does not mean you have to be friends with your ex. It's seen as the ultimate sign of maturity; no, we're no longer together, but we still care about each other...don't do that. It's just not worth it. Sure, there are exceptions, and someone will undoubtedly chime in that their ex is their best friend...but that's the exception and not the rule. Most of your exes need to be shipped off to an island where you won't have to see them everyday.
We can still be friends, right? If someone says that to you, of course the natural impulse is to say yes. No one wants to be considered bitter. That said, you have every right to say no. If someone hurt you, or your relationship was toxic, you do not owe your ex your friendship. It is okay to say no.
Humiliation. If your ex broke up with you, no matter how good your intentions are, when you run into them in person, you're going to feel humiliated. Especially if it was a messy breakup. It hurts to see the person you love in a new capacity. Especially if they're with someone new.
You're not over it. If you're not over your ex, why would you want to torture yourself by seeing them? Are you some kind of masochist? It's not going to make you feel better. It's going to make you feel worse.