Since those "weeds" grew there and removed nutrients from the soil why not just cover them with newspaper and a good mulch and let the soil bacteria digest them and put the nutrients they took out of the soil back?

Or you could remove as many as possible by beheading them with a loop hoe and picking them up. Get as many as you can without making yourself crazy. You might eliminate some runners and seed heads that way. Then mulch.

But why do all that work when covering those weeds with newspaper will kill them and put the nutrients the removed from the soil back into the soil and since those plants are now deprived of sunlight, something all plants need to grow, they will not grow and the seeds will not germinate.
Much less work covering the soil with newspaper and a good mulch to hold the paper in place as well as hide that paper.

what if I want to plant a fall garden? Can I just behead the weeds and till the rest under?

You could, although it is known that tilling soil brings up more "weed" seeds that can germiate and grow and need removal, or you could lay the newspaper and mulch down after the fall planting is in and growing.

pj, tilling is a good way to invigorate many weeds and grass that can continue growing from small pieces of roots or stems. Knowing what is growing there first would be necessary before giving advice about your potential success with any method. You're welcome to start a new discussion for your questions.

Could you reccomend a "good mulch"? I too have many weeds this year. I save all the leaves from our property, but the resulting leaf mulch is no where near enough to cover our garden. I did the newspaper/mulch technique but only had enough of our leaf mulch to do 2 rows. I am looking for free really inexpensive solutions, as most are. What are mulching options?

What about the leaves from your friends and neighbors?

Yep, do that as well as pick up bags of leaves on the way to and from work..but there is only so much space that I have to pile the leaves....unless I just pile it on the newspaper directly in the garden, not aged or broken down at all. Would that work? Or would I have to forego planting for a season?

Fall leaves heaped on the garden will be broken down for the most part in Spring, its fine to use them not aged either on your newspaper or without it. If the weather is good (and they are something like large leaf maple), I might run them over with the lawnmower to shred a bit, empty the clippings bag with its load of chopped leaves...or I may not and just put them on whole. We have more wind and rain here than outright cold, so I will sometimes add some finished compost or composted manure to help hold the leaves down until they start to decompose, add interest for the worms :)

For years I have picked up hundreds of bags of leaves others leave on the curb for disposal, shredded those, and made compost with some and used many for mulch on the planting beds in the fall. 6 to 8 inches of shredded leaves on the planting beds will last until about the first of August here helping suppress some "weed" growth, aiding in maintaining soil moisture, aiding in keeping the soil cooler, and adding organic matter to the soil.

So kimmsr, what do you do in the spring? Do you till it all in or do you just plant directly into the leaves?

Move the leaves aside a bit and plant and then move that mulch back. I have not tilled the planting beds I have in many years. The planting beds, once 6 feet wide and now 4, do not get any traffic on them that might compact the soil and the mulches help the Soil Food Web stay active and keep the soil in good tilth so tilling is not necessary.

If they are annuals mulch them or turn them under.
If they are perennials, then pull them or spray them.