This explanation might not be the greatest but...

Spoiler for justification for toppledLet's assume that the die is rolled. The die has two rotational degrees of freedom, but since it is symmetrical around its center, we can just consider one of them. If the die is rolled along its curved surface, it is easy to see that it is guaranteed to stay toppled. If the die is rolled after being rotated 90 degrees through the rotational axis orthogonal to the axis along which it is being rolled, such that the die alternates between toppled and standing until it stops, it is easy to see that the probability of it being topped is 50%. Here's where I take a speculative leap... The two aforementioned orientations are clearly the two extremes. For all the intermediate orientations in the 90 degrees interval between the two extremes, the probability of a toppled outcome should take all the intermediate values between 50% and 100%. Obviously, that would mean that toppled is the more probable outcome overall.

Spoiler for meanwhileIf the die is dropped, the probability of it landing on either side of its tipping point is 50%...

Spoiler for

..Most likely correct.Nice analysis!

I posted a similar problem on a science forum before..it is about the thick coin that has a head- tail-side probability of 1/3,i got a good reply-