To specifically answer the question - a T-test is a parametric test which requires a normal distribution. So it's not a good idea in context to this data. However, the non-parametric equivalent of the T-test is the Wilcoxon and thats absolutely perfect and described below together with the entire rationale for parametrics versus non-parametrics. Basically, I agree with your approach, except I would simply shift to non-parametric statistics.
I noted the response on Biostars from the link recommends DSeq2 and EdgeR. Both tests along with the additional test described below, along with the T-test, are used to identify whether each gene in the analysis is significantly up-, or down-regulated and can then be ranked. The underlying issue is the validity of the tests associated P-values.
Both DSeq2 and EdgeR have been notably criticised by Li et al 2022, in a paper entitled
Li et al 2022 Exaggerated false positives by popular differential expression methods when analyzing human population samples" Genome Biology: 79
Li et al (2022) needs to be read carefully before making a choice. Li et al 2022 is recommending using a non-parametric test, via Wilcoxon, which is implemented in the method NOIseq from the R-Cran Bioc library (described below). Genome Biology has a current impact factor of 18 (exceptional). I was told that one member of the DSeq2 team produced a later paper advocating non-parametrics methods.
The underlying issue is the dependency on a parametric distribution for both DSeq2 and EdgeR. Basically, the central issue is the validity of parametric statistics (assumed distribution) versus non-parametrics statics (no assumptions about the distribution).
In parametric statistics outside DE, i.e. in general, when these are used numerous tests are performed to assess the fit of the data set to the parametric distribution assumed. For example, for ANOVA it's the Fmax test. For a T-test there's a number of tests for the normal distribution. If the data set fails the test, a multitude of transformations are possible usually aimed at normalising the variance.
From this perspective the concern is no comparative tests were devised for DSeq2 or EdgeR given the age of the papers DSeq2 is from 2014 Love et al (2014). If there's no test how could the data use alternative transformations to fit the assumptions? Again this is unusual given the age of papers because such as tests would be expected.
The non-parametric method mentioned, NOISeq, is from the R/ Bioc package by Tarazona et al 2015, Nucleic Acids:43. This was the approach use by Li et al 2015 and compared against both DSeq2 and EdgeR. There are more recent examples, again I believe one of the DSeq2 team produced a non-parametric DEG as well.
The GitHub page for Li et al 2022, is https://github.com/xihuimeijing/DEGs_Analysis_FDR
I understand the criticism recently stated about Wilcoxon (non-parametrics in general) is their reduced power (response on Biostars). What happens is you get fewer significant 'hits' that are under or over-expressed genes. This is the basic trade-off between parametrics vs. non-parametrics. The key issue is if you've trapped your key results within a non-parametric test, that is a very solid result. If the equivalent results are trapped using EdgeR or DSeq2 there is always going to be question mark whether it's a false positive.
When to use EdgeR/DSeq2: if you are doing alternative statistical tests on the expression data which are directly comparable with under- and over-expression (e.g. ML and DL analysis) thats okay because we know Edge/DSeq2 gives false positives. When I've done this there's a 'centre ground' which can give conflicting result, its the central quartile type area, i.e. 25 - 75% region (to be honest it can be a bit outside that).
If you are singly dependent on EdgeR, DSeq2 or NOISeq it is better to be conservative in my opinion, i.e. fewer hits which you are more certain about.
Final point The final issue is simply how close to the DSeq2 and EdgeR data sets is the data you are working with, i.e. same species? What I would caution is these distributions have been applied broadly across species. The extent of the false positive rate must vary in my opinion.