蒲公英 is not a big deal. They wouldn't live over the winter. I usually don't pull them. Just mow them down, The key is mow them before they seed. When you see flowers, mow them. For the strongest one I would use weed killer like roundup to kill them.
蒲公英 grow faster then grass but don't take mowing very well.
Grass would take mowing and would out compete 蒲公英 if you put seeds, fertilizing and watering.
So if 蒲公英 is your only problem. You should buy grass seeds, fertilizer . rake the lawn off dead grass , expose the soil, spread the seeds and rake again make sure seeds are mixed in with soil, spread fertilizer then water.
Most lawn go bad is because the owner did not do the basic, seeding, fertilizing and wait to long to mow the lawn.
The grass is not 野草 是庄稼. lawn is not natural grass land, is farm field. Did you even seen natural grass land has only one kind of grass and so tightly grow together. The lawn is very unnatural environment. If you want it to look good, you have to feed the lawn.