Does your point remain valid if you take a realistic distribution over coin imperfections into account?

Possibly irrelevant calculation follows (do we have hide tags? Apparently not)

Suppose we have the simplest sort of deviation possible: let alpha be a small number

P(10 heads) = (1/2+alpha)^10

P(HTTHTHHHTH) = (1/2+alpha)^6*(1/2-alpha)^4

Remarkably (?)

dP(10 heads) /dalpha = 5/256 at alpha=0

dP(HTTHTHHHTH) /dalpha = 1/256 at alpha=0

It seems that simple coin deviations (which are by hypothesis the most probable) have a stronger influence on simple predictions such as P(10 heads) than on complicated predictions such as P(HTTHTHHHTH)