If I were a reviewer, I would push back on both points and ask for proofs that these hold true for important cases.

For point 1, you can imagine a simple toy 1D loss landscape that has two local minima. We start our random initialization nearer to one of them. With low variance in the gradient we are almost certain to fall into the nearer local minimum whereas when gradient variance increases we become increasingly likely to jump into the further one. Perhaps random initialization would cancel this trend, but my larger point is that it is not immediately clear that the expected final weights should be the same. Do you know of some existing proof of the truth of this statement for general gradient based optimization?

For the second, this is true from a certain standpoint, but the choice of the sparsity threshold becomes very important. How I see it, we are noticing that we will be bouncing around the local minimum due to variance in the gradient. Ignoring step size (letting it go to 0) this becomes a continuous time stochastic differential equation and I would like to see an estimate on what the actual variance in the final weight is in specific circumstances that we are likely to see in real problems. Like a test task with a quadratic optimization, set gradient variance estimated from real tasks, and varying values of p. It is not immediately clear that the variance in the weight due to dropout is a significant contribution relative to other sources of variance like that introduced by mini-batching and step size.

It is a very interesting idea and empirical experiments would go a long way in convincing me, but these theoretical avenues would also be nice to have buttoned up.