According to some videos I watched, weed burners only need to heat the weeds enough that you see a whiff of steam, not actually set them on fire.
Now, maybe that only applied to emerging weeds, not "adults".
I would have thought that, instead of pulling each flower as it bloomed, you would pull or chop each stem as it developed identifiable leaves.
(Listen to me make energetic suggestions: I have beds COVERED with weeds.)