I think I have part of it. Assume zero interest rate and T = 1. Then the call price C is
C = S.N(d1) – K.N(d2)
where S is underlying price, K is strike, and
d1 = ln(S/K)/V+V/2
d2 = d1 – V/2
d1 and d2 roughly represent the moneyness in terms of standard deviation, including the term V/2 which is added in d1, and subtracted in d2. Nd1 and Nd2 represent the moneyness in terms of probability. Note that the deeper in the money, the closer the probability gets to 1.
Now when S > K, it is easy to show that time value must be positive. Let X = 1-Nd1, and Y = 1-Nd2. Then
C = S(1-x) – K(1-Y) = S-K +Y-X = intrinsic value + Y – X
Since d1 is always a bit greater than d2 because of the V/2 term, it follows that Nd1 is closer to 1, and so Y>X.