I get a little paintbrush and paint the leaves of each dandelion with round-up - that ends up killing them but largely leaving other plants alone.
Yours is so much more.. tender though. Poor dandelions, but at least you made it personal!
tl;dr targeted herbicide is a much less evolutionarily selected-for offense, as opposed to hand cultivation which mimics attacks plants have evolved to survive for eons
Even if you do successfully get it out, it really is going to be more work than painting a weed killer on them.
https://images.wur.nl/digital/collection/coll13/id/676/rec/3