Yes, mowing will keep them at bay but not kill them. Pulling is a good way to go but will not kill them either. For paths and walkways use cheap white vinegar in a sprayer on a hot day. Vinegar will kill grass as well so do not use in yard.
I too am against chemicals but I do use spot weed killer on my lawn. The gallon sprayer has lasted me over 2 years.
Getting them under control and then staying on top of them is the best plan.
It sounds as if your lawn is in need of some serious work and sorry to say that is what it will take.
I am still mulling over the idea of turning my entire back yard into a veggy garden.
Good luck.
j
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