Sadly for the title, I think the word you were looking for is "gimbal," not "gyroscope." A gyro rotates stably on a single axis. Your cooker "tumbles" on three axes at once. A gyroscope specifically prevents tumbling.
It does indeed work! I've been meaning to instrument a food stimulant mass to determine how the chaos is effected my mass offset vs speed. I think due to the under-actuated nature of the system you're stuck with a balance between cg-offset, gravity and input rotation on the first axis
Yeah, it seems this free-rotating design would keep one side down if the center of gravity is not exatly on the intersection of all rotation axes. This might be worse than a common single-axis cooking.
I watched the short gif/video and I had the same thought. If you don't pierce the object/meat in a manner that its centre of gravity is on the very centre of this contraption, it would skew the spin and it affect its randomness in the movement.
But totally a fun project and cool topic to discuss on any BBQ
I built a contraption to slow cook in multiple axes and documented it here:
https://transistor-man.com/gyroscopic_gyros.html
Not only is it tasty, it's mesmerizing to watch. Feel free to copy the design for your own festivities.