Dear Farbod-
I have seen some papers that take that approach, or to report all results along with a venn diagram and focus on only those results that are in common for all methods.
However, I don't like either of those because each method makes different statistical assumptions about the data and handles outliers and biases in your data differently. If the data were ideal (i.e. from a simulation) then all three should give very similar answers and these approaches would be valid. However, with real data it is common that some of the assumptions of a model will not apply perfectly. For that reason, you will get different answers from each of the methods and some of the methods will contain more type I errors while others will contain more type II errors. Unfortunately, you can't tell just by looking at your lists of genes which are which.
For these reasons, what I did, was to explore the results of multiple methods, like you suggest, but then determine which method produced results that I was most confident in.
In the paper only those results are reported. I didn't use voom, but I compared DESeq2, edgeR, and EBSeq. At the "gene" level, I found that DESeq2 produced a very nice (symmetrical) MA plot and a list of genes that I could be very confident in (after exploring the data carefully). Unfortunately, using DESeq2, I lost a few interesting genes that were removed due to the outlier procedure, but it was hard to be confident in those genes anyway. I also used EBSeq to look at transcript-level DE. I was more comfortable with the Bayesian approach to isoform uncertainty taken by that method and the results complemented those of DESeq2 nicely. However, I think I would have found almost exactly the same results if I had used DESeq2 for both.
From what you described about your results, I would say that things are looking good for DESeq2. That is considered by its authors to be a very conservative algorithm and they recommend starting at a FDR cutoff of 0.1. I have seen papers that use 0.1 or 0.05 (what I used in the PLOS Pathogens paper) or more stringent cutoffs. You should run your data through the entire DESeq2 vignette to see how it looks compared to the sample data. This will help you decide whether it is meeting the assumptions of that statistical model. If it looks good after that, then you can have confidence in selecting DESeq2 for analysis. I would run it at different FDR cutoffs to see how that affects the genes that you have and choose a cutoff that you are comfortable with based on the numbers of genes and the expected false discoveries.
I hope that this helps.
Sincerely,
Ken