Maybe it won't matter much because at this point I only have empirical evidence, but I found t-SNE to outperform UMAP for hundreds of metagenomic datasets. Not sure why, but most people don't seem to know about this package:
https://github.com/pavlin-policar/openTSNE
Not only does it use by default the same initialization as outlined in that Nat. Biotech. comment, but it is multithreaded and therefore doesn't have "the scaling problem."
Makes pretty good animations as well.

Hi rtrende, what package do you use to run UMAP?
thanks
I've been running UMAP using Seurat, which uses the python umap-learn package
There is also the umap package in R (on CRAN).
The Bioconductor package
scateroffers convenience functions for both t-SNE and UMAP.Part of the issue with t-SNE is that you get different results each run, it doesn't scale well, and the "rigorous" improvements you mention require extra setup or aren't supported in most packages. If it's shown to be a real improvement, it will likely be adopted in time as people become more aware of it (as was/is the case for UMAP). Convenience often reigns supreme.