I didn't see any normalization being done (nor subsampeling) in the ANCOM code So, and you are most welcome to correct me if I'm wrong here, as I understand it - we assume ANCOM isn't sensitive to varying sampling depths (In the paper, they don't address sequencing depth issue).
Can you provide an explanation how ANCOM overcomes false discoveries originating from differences in #reads?
Now, including low abundant features in the test may also cause false negative inferences as the FDR correction being done on a much larger feature set, and deeper samples tend to have more of these low abundant features (features that you usually get rid off once your subsample the data). Are we sure this is a good idea to test for differential abundance on samples of uneven depth??
Concerning the normalization - ANCOM is self normalizing (see here) in the sense that the sequencing depths cancel out.
But a good sanity check, I would recommend running ANCOM on both rarefied and unrarefied data. If the above concept is truly working, both rarefied/unrarefied ANCOM results should give the same results (with the exception of a few low abundant taxa).
The false discovery control in general is a bit more broad though. Part of the procedure assumes that at least 2 taxa are changing (I personally think it is 50%, but that’s for later discussion), and it looks at all possible pairs log ratios.
By doing so, you avoid may of the issues associated with compositionality (since for a given log ratio, you are only focused on 2 species, random fluctuations in other species won’t impact the specific log ratio).
My paper here also highlights some of the false discovery issues associated with proportions.