I'm pretty sure that the NIH does do studies into efficacy of plants. This was after a ten second Kagi search: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3059459/. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3847409/
I don't know if that counts as "publicly funded medicine", but it's certainly a publicly funded group that studies medicine.