I am curious how this changes if we condition on it flipping in the air at least once. Can we think of this result as a mixture distribution of a fair 50/50 chance of it flips at least once, and a delta function that is 100% at the side it started on, if not flipped at all?

Seems likely it would change. Here's another way to think about it:

0 rotations is more likely than 1 rotation, since there is a wider range of rotation speeds that lead to exactly 0 rotations than to exactly 1. Similarly, 2 flips is more likely than 3, 4 is more than 5, and so on. So you're always biased towards an even number of flips and the starting side.

Take out the 0 case by your conditional, and you're left with 1 > 2, 3 > 4, 5 > 6, and so on, now biased towards an odd number and the non-starting side.

The paper requires that the coins flipped at least once to be counted.