Gradient boosting vs logistic regression, for boolean features
You are right that the models are equivalent in terms of the functions they can express, so with infinite training data and a function where the input variables don't interact with each other in any way they will both probably asymptotically approach the underlying joint probability distribution.  This would definitely not be true if your features were not all binary.

Gradient boosted stumps adds extra machinery that sounds like it is irrelevant to your task.  Logistic regression will efficiently  compute a maximum likelihood estimate assuming that all the inputs are independent.  I would go with logistic regression.