Benja: This breaks the implicit decision theoretic premise that your payoff depends only on the action you choose, not on the process you use to arrive at that choice

Correct! The next step in the argument, if you were going to formulate my timeless decision theory, is to describe a new class of games in which your payoff depends only on the type of decision that you make or on the types of decision that you make in different situations, being the person that you are. The former class includes Newcomb's Problem; the latter class further includes the conditional strategy of the Prisoner's Dilemma (in which the opponent doesn't just care whether you cooperate, but whether you cooperate conditional on their cooperation).

However, within this larger problem class, we don't care why you have the decision-type or strategy-type that you do - we don't care what ritual of cognition generates it - any more than Omega in Newcomb's Problem cares why you take only one box, so long as you do.

Though it may be important that the other player knows our strategy-type, which in turn may make it important that they know our ritual of cognition; and making your decision depend on your opponent's decision may require knowing their strategy-type, etc.