For my application I need just the translations and Euler angles. The range of poses is mechanically constrained so I don't have to worry about gimbal lock. But yeah, my limited understanding matches yours that other parameterizations are more useful in general contexts.
This post and interactive explanations have been on my backlog to read and internalize: https://thenumb.at/Exponential-Rotations/
(Also: Thanks for pointing out the typo, I just deployed a fix.)