Yes and no, in various senses. In terms of points upon the Earth, the collection of points within five miles of the North pole is exactly as likely to be hit as, say, the collection of points within five miles of Detroit, or the collection of points within five miles of some point on the equator. In this sense, there is no special anti-vertical skew, in that the North pole is treated symmetrically with any other point. However, in terms of the particular measure of verticality of latitude, yes, latitude will be skewed away from +90. Latitude will not be uniformly distributed. But this is actually the right thing.
The right notion of a uniform distribution of directions (i.e., of a random direction) is such that the probability of the direction falling within a certain region on some fixed sphere is proportional to the area of that region. [This is the unique distribution with the desirable property of rotation invariance: if region R can be transformed into region S by a rotation, then R and S will have the same probability of being landed on; you get the North pole, Detroit, equator disc equivalences from before, for example.] However, you’ll note that the surface area of the portion of a sphere between, say, +90 and +80 degrees latitude is smaller than the surface area of the portion between +80 and +70 degrees. And other such things. So, yes, there is a “skewing” in a sense, in that the distribution of latitudes will not be uniform. But this is the right skewing, in that it’s the one which corresponds to a uniform distribution of directions. As explained before, it’s only necessarily a skewing away from vertical if you interpret that strictly in terms of latitude (which is a poor measure for these purposes, for precisely the reason that more of the surface area of the Earth is caught up in less extreme latitudes); if you look at it in other ways, it doesn’t seem like a skewing away from vertical at all.