in your view, what is the role of initial conditions in determining (a) the "objective probability" and (b) the observed frequencies?

In a deterministic universe (about which I presume you to be talking because you are talking about initial conditions), the initial conditions determine the precise outcome (in complete detail), just as the outcome, in its turn, determines the initial conditions (i.e., given the deterministic laws and given the precise outcome, the initial conditions must be such-and-such). The precise outcome logically determines the observed frequency because the observed frequency is simply a high-level description of the precise outcome. So the initial conditions determine the observed frequency.

But the initial conditions do not determine the objective probability any more than the precise outcome determines the objective probability.

Probability can be applied both to the initial conditions and to the precise outcome. Just as we can classify all different possible precise outcomes as either "heads up" or "tails up" (ignoring "on edge" etc.), so can we also classify all possible initial conditions as either "producing an outcome of heads up" (call this Class A) or "producing an outcome of tails up". And just as we can talk about the probability that a flipped coin will belong to the class "heads up", so we can also talk about the probability that the initial condition of the flip will belong to Class A.