Is there a way to create simple diagnostic plots from the DESeqDataSet object, plots similar to simply plot(fit) where fit is an lm object? Basically plots similar to this https://data.library.virginia.edu/diagnostic-plots/. I've dug around the DESeqDataSet object, but can't find the information I need.

Short answer is no.

We do provide the mean values "mu" and the hat matrix diagonals "H" in assays(dds), so you could make your own GLM diagnostic plots if you want.

https://bioconductor.org/packages/release/bioc/vignettes/DESeq2/inst/doc/DESeq2.html#access-to-all-calculated-values