More like you are accidentally forced into arbitrary rotations with the tumbler, because of how moves correspond to inconsistent axes depending on the way you're turned.
Honestly I don't see that supposed failing as much of a downside really, because A) most of those extra rotations don't change the camera view in any meaningful way, they just roll it, B) you can typically rotate the object on top of the camera if you really need these rotations, and C) it makes control much more accurate.
I think the "turnable" approach would be fine if it included an extra roll control. Probably preferable due to the ease you point out; this overcomes the limit you point out.
My personal preference, and how I've set up my visualization engine (For chemistry vis projects and similar): Move the camera; not the object. FPV controls, plus a roll axis, with crouch/jump for up/down. Works best with a videogame controller, but FPS standard + Q/E is fine too! Full control, and intuitive if you've played those sorts of games. This is full 6 axis.
Honestly I don't see that supposed failing as much of a downside really, because A) most of those extra rotations don't change the camera view in any meaningful way, they just roll it, B) you can typically rotate the object on top of the camera if you really need these rotations, and C) it makes control much more accurate.