Getting Feed Aggregator to ignore certain incoming data
OK, so I'm considering deleting this but it was such a source of frustration for me I guess I'll leave it up in case someone else ever comes across this. This answer, to me, seems like more a workaround (mainly because I am still lacking a basic understanding of why it's happening) so if anyone else has a better answer or more information please do post your own answer.

It seems like you can't have complete control over how Views or Aggregator arranges elements from the XML feed. It also seems (from my fruitless Googling) that this isn't usually a problem. I don't really know if this is an issue with how the RSS feed we have to use is configured or if it's a Drupal thing but I know the answer is in gaining control over where and when the different XML elements appear.

Most people seem to use a module called Feeds to gain control over this type of thing (and do other stuff). From superficially looking into it, it seems like it could fix my problem but it creates a separate node for each incoming item, which is not ideal for my special case. The feed on the site I'm working on just keeps up with the latest 20 items and then discards them so I didn't want to have to deal with a separate node for each thing. 

I've decided to try a different module called Views RSS which sounds like it can simply solve my problem (here's some more info). I haven't tried it yet but it looks promising. Thanks to anyone who read this and to anyone who might be able to contribute in the future.