Probability predictions in random forest and bagging with aggregating not a vote count: seeking documentation/existing reserach

Typically, across an entire bagging or random forest ensemble, class probabilities are determined through a vote count method. Within that ensemble, each tree makes a class prediction based on the majority class in a terminal node. Let $$T$$ be the total number of trees and $$b_t$$ be the $$t$$-th tree in the ensemble. Let $$\mathbb{I}(b_t(x_i) = A)$$ be the indicator function that returns $$1$$ when $$b_t$$ predicts that observation $$x_i$$ belongs to class $$A$$. The probability of class $$A$$ for observation $$x_i$$ is calculated as:

$$\Pr(x_i = A) = \frac{1}{T} \sum_{t=1}^{T} \mathbb{I}(b_t(x_i) = A).$$

A different method is that probability predictions are computed for each $$b_t$$ then averaged across the ensemble. It appears this this method is used in RandomForestClassifier.predict in Python. Although I am an R user with randomForest. In maths, this is:

$$\Pr(x_i = A) = \frac{1}{T} \sum_{t=1}^{T} \Pr(b_t(x_i) = A).$$

My question is that I seeking evidence of the relative performance of each method. Ideally, a simulation study to assist with my understanding in this area. I have looked around but I cant quite find anything out there. My intuition from the limited reading is that probability predictions from each CART model are so poor that their aggregate is a poor estimate of the probability anyway.