Activation Phase, Page 7:
"Breaking Ties
When multiple ships have the same pilot skill value,
the player with initiative must activate all of his
ships with that pilot skill first."
Combat Phase, Page 10
If both players have ships with the same pilot skill
value, the player with initiative resolves
his ships’ combat steps first (see “Initiative” and
“Simultaneous Attack Rule” on page 16).
So that's the two of the four phases in the Core Rules where in both cases, only if the ships have the same pilot skill does it come down to a question of initiative. In the case of the other two phases, there's no discussion of how anything is resolved. To my mind, the fact that it's two phases that do specify how things are done, trumps the other two phases where no discussion is had.
I'm still unclear as to why we would think otherwise that in the case of a mirror match with dual Corran Horn's with different pilot skills would ignore that in favour of a pure initiative bid?
By choosing to cut only part of the Initiative wording as was done above, it changes the situation. However, in the full text on page 16, the second paragraph makes the statement that only when ships of the same pilot skill are activated does initiative come into play. If both pilots had the same PS, then I'd agree that initiative is the defining factor.
Page 16 (bolding mine)
Initiative
One player always has initiative, a distinction used
to resolve timing conflicts. Unless using the squad
building rules (see page 18), the Imperial player
has initiative. Initiative remains with a player and
does not change during the game.
When ships of equal pilot skill value are activated,
the player with initiative activates all of his ships
with that pilot skill value first. Then the opposing
player activates his ships. Initiative also applies
during the Combat phase; the player with initiative
resolves combat steps for his ships with that pilot
skill value before his opponent (see “Simultaneous
Attack Rule” above).
If multiple abilities resolve at the same time, the
player with initiative resolves his abilities first.
So even on the Initiative rules on page 16, it states clearly (to me) that it only enters play - regardless of phase here as it's a general rule - only if pilot skill is equal.
You realize you quoted yet ignored the relevant rules text, which I've emphasized for you. We are talking about an ability after all.