Eternal Decks · Board game rules
For stage E can only the player who gained the 3rd star complete the starting task cards
Quick answer
No — any player holding a Starting Task Card can complete it the moment the 3rd or 4th Star is gained by anyone, not only the player who gained that Star.
Details
The Starting Task Card rule states: "The moment you gain the 3rd or 4th Star, if you can reveal a 1/2/3/4/5 in your matching color you will complete the task." [p.41 — Task Cards]. This is different from the General Task Cards, which explicitly require "you must be the one that performs the matching task, not the other players" [p.41 — Task Cards]. Since no such restriction is stated for Starting Task Cards, the trigger ("gaining the 3rd or 4th Star") is a team-wide event — whichever player achieves it, all players check whether they can reveal a matching card to complete their own Starting Task.
Each player's Starting Task Card is tied to their player color, and it can be completed using Number Cards or Two-Color Cards of the matching color and value, but not Rare Cards [p.41 — Task Cards]. Once the 4th Star is gained, this task can no longer be completed [p.41 — Task Cards].
Watch out: In Stage E+ (Advanced Mode), the rule changes significantly — the Starting Task Card can only be completed once the 4th Star is gained and all hands are revealed, and every player must be able to complete it at that exact moment or the group immediately loses [p.53 — Starting Tasks and Winning].