Adaptive vs Bernoulli sampling
I do not quite understand the question because you do not define some of your paremters (what are $p$ and $q$?). But let me try to clarify things. Let $N$ be the true number of distinct items.

First Flajolet's guarantee for Wegman's algorithm is that, if the space used is $n$ words, then the estimate $X$ produced by the algorithm satisfies 
$$
\sqrt{\mathbb{E}[(X - N)^2]} \leq \frac{1.2}{\sqrt{n}}N.
$$

Your suggestion is to sample at a rate $p$ from the $N$ different distinct items. I.e., you make a pass over the sequence of item identifiers and you keep each identifier in memory with probability $p$. Let the number of items stored in memory be $n$. Then the expected value of $n$ is $pN$, and the variance of $n$ is $p(1-p)N$, so we have
$$
\sqrt{\mathbb{E}\left[\left(\frac{n}{p}-N\right)^2\right]} =\sqrt{\frac{1-p}{pN}}N = \sqrt{\frac{1-p}{\mathbb{E}[n]}}N.
$$

So you are right that this is in some sense comparable with the guarantees for adaptive sampling (as well as those for the Flajolet-Martin probabilistic counting algorithm). However, the issue is that without knowing $N$, you do not know ahead of time what your space complexity $n$ will be, even in expectation. Or to put it another way, you do not know what to set $p$ to in order to make the space complexity acceptable. 

A natural fix is to start with some sample rate $p$, say $p = 1$, and once your memory fills up, you lower $p$. This is essentially the adaptive sampling algorithm.

The simple sampling algorithm you suggest can be useful to refine a rough estimate of $N$. For example, you can use adaptive sampling or Flajolet-Martin to get a constant factor approximation to $N$, and then use that approximation to set $p$ in the Bernoulli sampling strategy. See this nice paper by Daniel Kane, Jelani Nelson, and David Woodruff, where they use this idea and a lot more to get an optimal space algorithm for the distinct items problem. The paper also gives references to much of the more important work on this problem.