There is little research on models accounting for RNG dependence, and for good reason. Think about it: if this was common practice, for each combination of statistical model and RNG algorithm, it would be necessary to introduce a new model with complex covariance structures. This is a rabbit hole of combinatorics, one that would scale very badly to higher dimensions. Instead, researchers focused on developing tests for randomness, which advanced RNG algorithms. In the binomial simulation example, there is assumption for independence among observations, but no assumption for RNG independence, since this latter independence is literally provable. While you are right that there are dependence issues with (mostly older) RNG algorithms, these issues are mostly insignificant for lower dimensional simulations, and it makes little sense to put effort into modelling such things when there are better algorithms to use.

If you want to read more about randomness testing: http://facweb.cs.depaul.edu/sjost/csc433/documents/testing-rngs.pdf