I have Self Heal too! I'm growing very fond if it, if it gets into a place with bare soil which is reasonably fertile and partly shaded it grows much bigger, more attractive than many plants. At first I thought like many people do, it spoils the 'lawn' but I soon learnt that I don't have a 'lawn', I have meadow grass which will support many wildflowers if given the chance. I took over the garden with a good 3" of moss (which I like but it chokes everything else and dies in dry weather). I tried to rid the grass of moss by mowing to the soil, then fed the grass which worked for a season then the soil was once more leached of nutrients as it's sandy and acidic.
White Clover started to take hold, I let it spread and it fed the soil as it produces nitrogenous nodules on the roots. Now the grass gets one cut when it starts to grow well and maybe another later nearer winer, I let the rest do it's thing and I'm getting many different wildflowers. I walk around the beds a lot taking photos of insects so I have a beaten path. My neighbours cut their grass to the bone, with the current heat we are getting it's dying while mine may be well trodden and looking a bit less lush it is not suffering.
Dandelions are great in the right place, I have them growing on the south side of a hedge and they provide nectar and pollen for many early bees. They are better to look at that nothing.
Save your energy, do what suits you and enjoy the rewards which far outweigh a sterile garden. You are already doing that, congratulations!
