Within and between sample count normalization
The approach you are describing seems very strange.
Crucially, the Vignette for DESeq2 states that the model only works correctly with unnormalized counts as input:

It is important to provide count matrices as input for DESeq2’s
statistical model (Love, Huber, and Anders 2014) to hold, as only the
count values allow assessing the measurement precision correctly. The
DESeq2 model internally corrects for library size, so transformed or
normalized values such as counts scaled by library size should not be
used as input.

Normalized counts like FPKM are not supported and should be converted to counts using the appropriate import funcion.
I don't really understand the use of the normalization factors for quality control because the number of reads in a sample does not correlate well with the quality. Maybe you could link to the paper in case there is something hidden in the experimental design that would explain this.
If you are interested in quality control a useful metric in my experience is the 3' sequencing bias that can be calculated using, for example, RNA-SeQC.