My answer assumes that you're asking about having IMUs measure angular acceleration instead of angular rate; if meant they should measure angular acceleration in addition to angular rate, then I apologize for misunderstanding you.
However, having available the angular acceleration would be useful for a more precise kinematic model when integrating to find the pose.
I'm not sure where you're getting that information. When integrating to find the attitude, the angular rate is much more useful than the angular acceleration.
An attitude indicator essentially just takes the angular rate and integrates it. However, there are two sources of error here: errors from the sensor, and the fact that the direction of gravity changes as the earth rotates and as you move across the surface. But both of these errors can be corrected by mixing proper acceleration into the calculation.
On the other hand, if you only had access to angular acceleration data, you'd need to integrate twice to find the attitude: once to come up with an estimate of the angular rate and again to come up with an estimate of the attitude. (Why would you want to have to integrate to come up with an estimate of the angular rate, instead of just measuring the angular rate?)
This would introduce a third source of error into the attitude indicator: errors that result from accumulated errors in the angular rate estimate. I think these errors would be much more difficult to correct.
I think that the biggest problem is that in a coordinated turn, the angular acceleration is 0 and the lateral proper acceleration is also 0, so without measuring angular rate, an IMU would be unable to distinguish a coordinated turn from straight-and-level flight! That's a critical flaw.
In short, you need a reliable measurement of angular rate, and that can't be obtained from a measurement of angular acceleration.