Dandelions Re-Grow from Root Pieces?

I don't mind dandelions and have let them grow pretty much wherever they sprout. I like the fact their roots go deep and break up hard soil, and they grow where many other plants won't.

However, there is a section of the back yard where they are getting out of control.

I am now wondering, what is the easiest method to clear out this plot?
(Other plants are now planned for that area. )

If I keep the plants mowed to the ground, I suppose that would prevent them from flowering and make the job of weeding easier next year. Would it kill the plants too? Or do they regrow?

I am wondering if they behave in any way like Comfrey, which regrows from pieces of root.

Or do dandelion plants have to be pulled up by the root?
If so, how much of the root has to be dug up to kill the plant?

You definitely need to remove all of the taproot. Any section much thicker than, say, the wire of a USB charging cable has the potential to grow back.

Two things come to mind. The dandelions are growing because there is compacted soil in the area. If you could de-compact the soil, the dandelions would not like it as much and would not grow as much.

Also, chickens and many herbivores like dandelions more than grass. If you had access to some animals, they might be able to help by keeping the top "mowed" for you.

Not certain why anyone would intentionally want to kill off a dandelion patch.  So much good food there between the blossoms, greens, and roots.  Your liver will love you if you just incorporate what is “out of hand” into your diet.

Dandelions will flower tight to the ground, out of reach of the mower blades, if cut at the right/wrong time. If you are planning a garden, a layer of large leaves, paper, or cardboard, and a few inches thick pile of leaves/woodchips will likely suppress most of them. The thicker the mulching the better, and whatever grows back are usually easy to harvest.. and/or you could also loosen up the soil with a garden fork or broadfork and harvest the roots as you go.
If you aren't in a rush, you could let them flower for a while before they start making seeds, then collect the flowers for a recipe. At this point they are putting a lot of energy above ground so it would likely be a good time to do whatever form of suppressing you choose.

You can "sheet mulch" the area in question and plant your desired plants directly into the sheet mulch while it is doing its job (the undesired plants become food for your new plants simultaneously)