I prefer to leave as many roots in the ground as possible because it adds organic matter to the soil, loosens the soil, feed the worms & other soil critters, plus helps aerate the soil. Here's some useful links.
Like Mike said, I leave the roots in the ground for extra nutrients for the soil! It will just get better with age.
I cut down the plants at the base with pruners and depending on their size I will either just leave it there on top as mulch or stuff it into my compost bin. I'm in the suburbs right now so I use a metal trash bin for my compost.
When you replant, depending on what your mulch material is and what you are planting you may or may not have to move the mulch aside.
For my own garden I pretty much exclusively mulch with wood chips since I can get them free so I have to push it aside in order to plant new seeds. Saplings are not strong enough to push aside a heavy wood chip piece. If you have straw or some other light material as mulch you may be able to directly plant certain seeds that have strong saplings like beans. But it all depends.
And of course it is different for transplants! Push the mulch aside, transplant the plant, and put the mulch back within reason. I'll try to put the thinner more decomposed mulch that you should be able to find underneath the top layer by now directly around the stems.
My biggest challenge here is being in hot hot Texas and wanting to push the mulch aside for direct seeding which then leaves the soil to get dry way too fast and I've become spoiled from the mulch and will forget to water those areas. Oops.
I'm not the most experienced but those are conclusions I have come to after my two years of vegetable gardening :) (Yes, Covid got me into permaculture, yay!)
Hill Country Dream: Land Acquired > Home plans driving me crazy (still) > food forest pending
"no dig" systems generally require regular topping with mulch. The weed suppression is due to the thickness of mulch acting as a barrier to weed seeds that germinate. Mulch rots down a huge amount each year - good as it feeds the soil - so will need topping up. In the areas where I use no dig/mulch I typically put down 6 to 8 inches of woodchips each year. My experience has been that this is more important that carefully cleaning/preparing the beds. After that it is easy to spot weed the few things that do go through.
I use deep mulch like this round berry and fruit bushes. My annuals get a more typical light cultivation and topping with some fertile compost from the chickens.
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