On the (rather strange) notion of strategies in repeated games
Your proposed notion of what a strategy should be in repeated game can be adapted to all extensive form games has been called a plan of action by Ariel Rubinstein, who discusses this and related issues in the paper

Rubinstein, Ariel. &quot;Comments on the interpretation of game theory.&quot;
Econometrica: Journal of the Econometric Society (1991): 909-924.

Some concepts become harder to define, such as backward induction in games of perfect information. However, one can still do pretty much all of game theory. The book &quot; Rationality in extensive form games&quot; by Andres Perea uses the definition as a plan of action throughout.