Boyfriend is best friends with his ex girlfriend - should I be worried?

I've been dating someone for almost a year now, and he's the type of guy who has a ton of friends who are girls. He says girls are just easier to talk to. I don't have too much problem with it, but I get a little insecure that his best friend is his ex.

They have known each other for 6 years or so, dated for the first 2. He told me he hasn't been romantic with her for years. But from what I hear, I believe she still likes him.

When I ask the reasons why they are not together, but why they are best friends, he says:
They're not together because - she is lazy, sleeps through constantly even when they have made plans, has no ambition, lives off of parents money, can't keep a job, slightly unstable.
But they're best friends because - she's a fun girl who is a good friend and caring, he knows her more than anybody else and she knows him more than anybody else.

A few months back (while him and I were dating), they had a falling out because things were said in that she wanted him to marry her because of the best friend relationship they have and seems like she has never lost feelings for him. He told her no, as in he doesn't see her as someone he could marry unless she changes herself and that he doesn't even want to get married right now.

Recently, they have started communicating with each other again (I believe she reached out); he said they're trying to be friends again, but not sure. I understand how hard it is to lose a best friend so I told him that I hope things go well. He did reassure me that he is not going to "be" with her because of the way she is and he doesn't want to.

He has been very honest and open with me about her and their relationship since the beginning, and I'd like to be supportive (and not one of those crazy girlfriends who says he can't see his friends anymore).
But since this one is also an ex (who recently wanted him to marry her), I do get insecure sometimes.
Boyfriend is best friends with his ex girlfriend - should I be worried?
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