And this is another reason why Harry should not tell just anyone about the True Patronus - it would drastically dilute his authority over the Dementors and his ability to broker a unilateral deal superceding the one they've got with the Ministry.

Definitely. Don't tell enemies about your strengths unless you are positioning yourself to be intimidating. (Even then it is better to have the enemies believing you have strengths that you don't have while still being unaware of your actual strengths.)

Playing on the fact that Harry doesn't believe in souls while most wizards and presumably the dementors do.

Well, failing that it comes down to the real fact that the dementors do assign value to doing a kissy-suck thing to humans that effectively destroys the human.