I spent some time play-testing my game with the rules I posted earlier. This is what I found.
1. As I had suspected, trying to force opponent's pieces to collide with and destroy each other is pretty close to impossible. Neither I nor my friend I was playing with were ever able to do it. However, it is very possible to force two of your own pieces to collide this way, or collide one of your pieces with an opponent. I think it would still work to have this action be a point of interest in the game.
2. With the current rules, pieces are destroyed when they collide head-on. The big problem with this is that the board starts to clear up pretty quickly, and the game becomes much less interesting the less pieces there are on the board. I think it'll be more interesting if all the pieces stay on the board throughout the game, and I've got some ideas about what should happen when pieces collide, and how to get a sense of progression from the start to the end of the game.
3. As a player, you don't really get any sense that you have control over what's happening. This is by far the biggest problem I encountered. Even though nothing that happens is random, it's extremely difficult to grasp what the consequences of your moves are, so you sort of feel like you're just randomly moving pieces and waiting for something to happen.
For my opponent, this was partly because he couldn't really grasp how pieces wrapped around the edges of the board (whenever a piece moved off of the board, I had to tell him where it went). We were also playing in my Illustrator file, which was a pain in the ass. So I'm going to mark the board so that it's hopefully clearer how pieces move, and then I'm going to actually print it out, so it can be easier to play.
But, all that aside, I knew all the rules and I knew how the pieces moved, and I still didn't feel like I had much control over what was happening on the board. Solving this might just be a matter of simplifying the rules for movement (I've got some ideas on that as well), or it might take some serious rethinking of the central mechanic. I'm hoping for the former.
4. To end on a high note: My friend assured me that even though he didn't really feel in control, moving the pieces was still exciting for him. He enjoyed the way the pieces move and interact. That's exciting for me, because that's basically what I was building the game around. If I have to scrap everything else, but I get to keep that. then that's all right with me.