When it comes to my gardens, I have various weeds with various methods of dealing with them depending on their growth pattern, their seeding potential, and, quite frankly, whether I like them or not for other reasons in that particular location or in general.
I tend to like dandelions a lot (for food, and for worm/microbe habitat, and because I don't feel that they compete), so I try to work with them, letting the grow big and for several years, unless they are in an area that I want to grow another tap rooted plant like carrots which they do seem to compete with. I do sometimes chop them to let another plant have more light, like in my garlic patch in the spring when the garlic is small.
This is not at all the case for quack grass, timothy grass, (actually any grass), thistles, daisies, chickweed, and hedge nettle. These I have zero tolerance on. They spread via roots and tend to seed heavily if allowed. If I see them, and I have the moment to spare, they come out with as much roots as I can take-hopefully all of the roots. Generally I will mulch over the mess that I've made, since the exposed soil from the upturning as the plant was pulled is bound to have a lot of weed seeds (which perpetuates the problem). I have tried to chop and drop some of those tenacious spreaders, but they tend to take advantage of my weekness/laziness and come back to haunt me, and in short order.
My only success with chopping and dropping those types of plants, though, was to chop them down, plant potatoes and mulch heavily. The daisies and hedge nettles and chickweed couldn't take that. I still weeded out the grasses and thistles; they are the worst of my problems, but I knew that and weeded them before I chopped the rest.
There are others, like plantain, wild mustard and lambs quarters that I leave because I like to have them around for food and medicine. I might still chop and drop them, but I never pull them unless I think they are totally in the way, and then only if I am putting in seeds or transplants. The plant goes right where it grew, if it's a plant I like, otherwise it goes out of the garden, for
compost, hugulkultur nutrients, or mulch under dense trees.