Consider games played with a set $S$ where the two players play elements of $S$ in turns for time $\omega$, "and then" the winner is chosen according to some pre-determined payoff set $P \subseteq S^\omega$. The axiom of determinacy says that such games of a certain type are determined (one of the players has a winning strategy). However, $ZF$ axioms are inconsistent with the statement that all such games are determined.
The wiki page for the axiom mentions a reason to believe in it, based on infinitary logic. I find the reason compelling, but I don't see it as restricted to a certain type of these games. Is there a way to have total determinacy? I mean a reasonable set of axioms of set theory that includes it, or perhaps some completely different system.
If not, how to convince myself of the essential wrongness of unrestricted determinacy? I know that over $ZF$ it implies the axiom of choice, and then we can construct an undetermined game with uncountable transfinite induction. Somehow this isn't enough for me to shake out the intial intuition. Also, it wouldn't work without the unobvious axiom of the powerset (but then using determinacy would be hard since only countable payoff sets could be constructed).