>> the big weeds pull out easily and seemed to shade and prevent a lot of new seedlings
YES! My newest omnipresent weed is an undistinguished Snapdragon someone gave me. She said "this is easy to grow, in fact, it will spread without any work".
Well, the work comes when birds drop the seeds in every raised bed you own. These snaps grew fast and thick, and no other weed has a chance to compete. They do pull out easily, and a lot of root comes with them.
