need help understanding the benefit of score function estimator

The score function estimator a.k.a REINFORCE policy gradient in reinforcement learning is (from http://blog.shakirm.com/2015/11/machine-learning-trick-of-the-day-5-log-derivative-trick/):

$$$$\nabla_{\theta}\mathbb{E}_{p(z;\theta)}[f(z)] = \mathbb{E}_{p(z,\theta)}[f(z)\nabla_{\theta}\log p(z;\theta)]$$$$

The literature on this equation and its derivation is rich. However, I can't yet find any explanation for:

1) If we can compute the first expectation exactly (as function of $$\theta$$), is it ok to take its derivative?

2) Is it ok to sample from the first expectation? In the above link, the author wrote: $$\nabla_{\theta}\mathbb{E}_{p(z;\theta)}[f(z)] = \int \nabla_{\theta}p(z,\theta)f(z)dz$$ where the derivative sign has been moved inside the integral sign (which the author wrote is (mathematically) valid). If then, can we estimate the gradient with: $$\frac{1}{S}\sum_{s=1}^S f(z^{(s)})\nabla_{\theta}p(z;\theta)\quad z^{s}\sim p(z)$$ since the above is unambiguously different from score function estimator (where there derivative of the log of $$p(z)$$ is taken), I guess my question should be why can't we do that?

Thank you!

2) We can't estimate the gradient with $$$$\frac{1}{S}\sum_{s=1}^S f(z^{(s)})\nabla_{\theta}p(z;\theta)\quad z^{s}\sim p(z)$$$$ because this is an estimator for an expectation. An expectation when written as an integral looks like $$$$\mathbb{E}_{p(z)}[f(z)] = \int p(z) f(z) dz$$$$ The integral we're trying to estimate is $$$$\int \nabla_{\theta}p(z,\theta)f(z)dz$$$$ which is not of the form $$\int p(z) f(z) dz$$ where $$p(z)$$ is a probability distribution function. This is because: just because $$p(z, \theta)$$ is a probability distribution function, this doesn't mean $$\nabla_\theta p(z, \theta)$$ is a probability distribution function.
Your given summation is actually estimating $$$$\mathbb{E}_{p(z)} [f(z) \nabla_\theta p(z; \theta)] \neq \nabla_{\theta}\mathbb{E}_{p(z;\theta)}[f(z)]$$$$