Some horses are just easy keepers and stay fat on very little. A true grass belly (I don't think it looks like your horse has one) is not really "fat". I've seen horses with grass bellies that in reality they're not fat at all. Their butts won't be nice and round and they can still have ribs and back bone showing above their big bellies. The big belly is from being full of undisgested hay/grass. Your horse doesn't look bad at all where her weight is concerned from what I see in the pictures. She just may be a little soft.
As far as a horse staying in shape on their own. I was talking about one staying in shape enough to do strenuous work once a month or so. I was surprised by the amount of people at the competitions I went to who didn't ride much and claimed thier horse was in shape is what I was talking about. I had those people sometime tell me I "over" conditioned my mare.
Becky