I don’t think you are understanding initiative correctly. Remember not to bring any ‘baggage’ from other A&A games into BOTB. You’ll end up reading things into the rules that aren’t there.
On the first turn only the Axis makes attacks. There is no roll.
On each subsequent turn (Turn 2 onward) roll initiative to decide which side gets first choice of hex to attack from. Once the initiative winning player chooses a hex, pays the supply or supplies (if attacking multiple hexes,) and resolves the attack- then the other player chooses a hex to attack from, pays the supply and resolves the attack. Each side alternates choosing a hex to attack from.
A hex may only attack once per turn.
The players continue alternating until either all hexes have attacked (you keep track of this by flipping the required supplies to the arrow side into the attacking hex pointing where they are attacking) or both players pass consecutively (when it is your turn to choose you are not required to declare an attack.)
You then proceed to movement and reinforcement and cleaning up the spent supplies and moving the front line markers.
You roll each turn to see who gets to choose the first attacking hex. If the Axis wins on turn 2 it does not automatically give the Allies first choice on turn 3.
Each turn each hex that is in supply may attack.
It is possible that one side will never get first choice.
In air turns the initiative winner gets to choose who will place their aircraft first. It is usually an advantage to be able to react to the opposing air force.