From my perspective now, I expect the reality to be the winning case 50% of the time because we are told this as part of the question: Omega is trustworthy and said it tossed a fair coin. In the possible futures where such an event could happen, 50% of the time my strategy would have paid off to a greater degree than it would lose the other 50% of the time. If omega did not toss a fair coin, then the situation is different, and my choice would be too.

There is no value in being the kind of person who globally optimizes because of the expectation to win on average.

There is no value in being such a person if they happen to lose, but that's like saying there's no value in being a person who avoids bets that lose on average by only posing the 1 in several million time they would have won the lottery. On average they'll come out ahead, just not in the specific situation that was described.