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What's worse, we've caught the receiever reporting a few centimeter precision while being meters off.

All other issues aside, this is a problem. Your sensor is lying to the filter about its accuracy.

I'd really need to see your config and some sample sensor messages to comment much more, but just so we're on the same page, you don't have to fuse yaw from the mag into your EKF state estimate. navsat_transform_node just needs it to obtain your pose in the UTM frame. Having said that, if you lack an absolute yaw source (i.e., if you are just using wheel odom and IMU and your heading estimate is drifting without bound), then you'll get into cases where your robot will appear to be facing in a direction that differs from its direction of motion.

One thing I've wanted to do for a long time is obtain heading by differentiating GPS positions. That would eliminate the need for mag data, though it would require the robot to move before a heading was generated.