I have changed some of my "asides" to standard posts because of this quirk mentioned by the OP. (I use Coraline, not Twenty-Ten.) I realized that when found in a search or using a tag or category link, the asides did not display the subject/title header, so that wasn't clickable, and the more tag also wasn't showing up, so only a tiny part of the post showed (and no pictures, thought the pics were available in "standard" posts appearing in the search), and there was no way to get to the full article.
Fortunately, I've only been using asides for a very short time, so this wasn't a huge problem for me. I just wish that the documentation on asides was clearer about how making a post an aside would affect it being viewed in wp searches, etc. If there is more info about that somewhere, I didn't find it -- but I'd love a link if anyone has one to more info!