on each 360 degree spin, it will be facing up for the first half of the spin.
Yes, and that’s decisive if the coin is stopped during that phase. It will be facing down for the second half of the spin, which is decisive if the coin is stopped in that phase instead.
Since coins can spin with different speeds and can be stopped after different periods of time, this should be somewhat random, once it’s spinning.
I think I got your point that on average, the coin is facing more up than down, since it started facing up. But I think that’s only relevant if the coin is stopped in that initial phase, before making at least half a spin.
Wait, are we approaching the same argument from different sides? If we assume a random distribution of spinning speeds and a random distribution of toss durations. Then there will be some coins which experience very slow rotation, and which are also caught early enough that they don’t complete even half a spin. These have to face up.
All the other combinations of spin and toss produce random faces.