the oldest and simplest form of plant breeding is selection---find a plant with some characteristic you like, keep selecting for that characteristic in future generations and eventually you get a new strain or variety that breeds true for the traits you've selected. one could also select against certain characteristics by making sure that those characteristics are not permitted to continue into future generations by culling off-type plants or otherwise not allowing them to breed.
i see weeding as a type of selection---especially with the way i do things in the garden these days. when i started gardening i tried to keep the garden free of all weeds. a lost cause of course and probably not a great way to go about things anyway. i've come to believe it's good to have some weeds around. it confuses the pests, helps keep a little more moisture in the soil, gives the beneficial critters some habitat, and some weeds are actually really good food. so i've been practicing selective weeding and it seems to be working out just fine. now i just manage the weed population, i don't try to eradicate them. i keep things clean enough that there isn't too much competition for the garden varieties to deal with and i encourage the edible weeds to grow and reseed themselves.
i've got stinging nettles, chickweed, and rocket growing in amongst the kale, pac choi, beans, etc. the weeds are just as tasty and nutritious (maybe more so) as anything i've planted and i didn't have the work of planting them. it's a pretty good deal and it makes meals a bit more interesting around here.