Am I correct to assume, that if the axis has a victory chit, that only the axis can gain victory points for controlling that objective. And, that if the allies control that axis objective, the only thing that it provides them is denying the axis from receiving the victory points?
In other words, neither side actually gains victory points for controlling the other sides specific objective hexes.
No, this is not true. Both sides benefit from the control of secret objective. This means that if enemy controls your secret objective then you must try to take control it or else the opponent will receive points at the end of game. If the opponent didn't receive points then in fact you would have no motivation to fight for control of the objective.
And for clarity's sake, if you have a Secret Objective chit, you get NO points for controlling that objective until that chit is revealed at the end of the game. (I don't remember if Pacific has the event which reveals one as Europe does...)