It's definitely interesting, and something we've though about, but there are some real challenges that make it not as simple as people may think:
So first of all, trim runaways commanded by autopilots kill, that much we all know from reading NTSB/news reports. Our autopilot is designed to mitigate that possibility in a couple of ways. First, the autopilot only pushes on the control surface actuators directly, just like your fingertips do on the stick. To limit the autopilot's ability to get you into trouble in a hurry, the servo has limited control authority, and so once you get somewhat out of trim, the servo starts to "slip" when it can't do what it wants to against the forces that are required to overcome an out of trim condition. Additionally, there's the trim annunciation, which tells you when you ARE out of trim but DO still have enough autopilot control authority such that the servo isn't slipping.
In today's methodology, if you're being asked for some trim by the AP, you'll tap the trim button or spin the wheel some. You shouldn't have to do this often, by the way. Every now and then in cruise, perhaps at the start and and of climbs/descents. You, the pilot, are part of the AP control system, and this is important: It's very unlikely that, if, somehow, the autopilot trim detection system failed, that you'd just keep applying trim mindlessly. You KNOW (because you engaged the AP relatively in trim) that it should be small adjustments.
An system that auto-drove the trim, however, doesn't have the same ability to detect when it's being asked for excessive trim when its sensors fail. Yes, the technically oriented among you can think of ways to characterize to mitigate this by characterizing the trim required, or limiting the amount of trim that can ever be put in by the autopilot to some subset of the full trim range, but, in the end, if you miss a failure case, and the system runs away with the trim though some catastrophic failure, it can be a bad day. A very out of trim airplane can create forces that are between hard and impossible to physically overcome without first reversing the out of trim condition.
Anyway, our view is - once you automate the trim control, there's a whole other level of redundancy and failsafeing that needs to be considered, and it gets complicated quickly.