chris is right. you have to break the cycle which means killing the weeds before they produce seed.
you have a lot of weeds because they came with the topsoil. we had a load of topsoil brought in to my mom's property. never had a problem with johnson grass. this load of soil was loaded with seeds. i have done my best to keep after it. anybody who has battled johnson grass knows it is difficult to kill with selective herbicide and has a very deep and rhizome based root system much like bermuda. yes, roundup will do it in, but only at the expense of whatever it is mixed in with. it is imperative that you watch closely when new topsoil is brought in and nip any newcomers in the bud.
yes, native weeds do some beneficial things for the soil. that doesn't mean you want them in the garden. they are first and foremost competitors and very efficient ones if they are native to your climate. sometimes we harness weeds and plant them on purpose to improve the soils. clover/vetch/cowpeas/soybeans etc. all fix nitrogen to the soil. they are beneficial if grown at the right time, but if allowed to go to seed each season, you will continue to grow that same crop and it will be very difficult to grow anything else. you have to break the cycle before they germinate by harvesting/cutting or destroying the root system.
if you don't continue each year to introduce new topsoil that isn't high heat composted and if you have decent barriers (fences, natural isolation, etc.) so new seeds don't blow in or get carried in by birds and others, you can expect the weed problem to diminish each season if you keep tilling or plowing or hoeing before they go to seed.
keep in mind that different weeds flower and seed at different times. the key is to watch for flowering or "bolting". once the flowers come out or the seed stem shoots up, it's time to do them in. seeds are eminent!
yes, organic is more work, but is worth it in my opinion. stick with it. it will pay off once you get a handle on it.
also, watch your mulch. using a hay mulch is akin to sowing hay seed (wheat, oat, rye, etc...) and there is no quicker way to get an unwanted crop growing...
amp