Fallow year for front yard veggie garden - help!

TL;DR -

Need the easiest, quickest possible solution for my front yard veggie garden.
Is it too late (edge of zone 5 and 6) to plant a legume or some cover crop for a fallow year? Would it be too late at the beginning of June?
As an alternative, is black plastic or cardboard a good way to let a garden go fallow?


DETAILS -

So here's the scenario. I have a front yard veggie garden consisting of 5 raised beds totaling around 200 square feet. Surrounded by chicken wire.

I originally set it up over the course of a few years almost 10 years ago. I basically dug out the grass by hand and then had the city deliver several yards of finished compost which I used to fill the raised beds and worked it into what little topsoil-ish stuff was there. It was incredibly rich and thick. Since then I haven't added a ton of nutrients but I do move things around every year and I work a good amount of tree leaves in most years.

At this point the raised beds are crumbling, in fact most of the inner walls have caved and I've removed them. So the chicken wire surrounds the outer perimeters of the 2 areas. At this point it is more like I have 2 raised beds than 5 since it is all sort of merging.

Life is happening in a big way right now, the last 2 years my garden has felt like a chore and has been unsuccessful to boot because I am not giving it attention, the soil is probably depleted, etc. So this year I am giving myself a pass. Phew.

Right now, it is overgrown with dandelions, grass, some overwintered flowering kale, tiny maple trees, and a ton of other weeds and things.

At the very least, I need to clear out the weeds and cover everything up in some way.

I am thinking I should plant some sort of cover crop, but I don't know much about these things and it is pretty late in the season...

It may be another week or two before I can get the weeds pulled (even half-assed) and toss some seed down. Is it too late? I am on the border of zone 5 and 6.

I've also heard about just covering the whole thing in black plastic, or cardboard, or something - does this help? Hurt? Kill weeds but otherwise a benign idea? I just don't know....

I also don't think I'll be able to deal with getting wood chips or compost delivered or anything. (I do have access to a ton of clean cardboard).

Basically I need to make this as easy as possible, and hopefully not totally unattractive. Any suggestions?

Also, not sure if it makes a difference, but I have been wanting to pull down the chicken wire, tear out all the raised bed wood, and start over in the ground without raised beds. *If* that happens, it will be in the fall. So basically any success in getting rid of weeds and improving the soil will be of great benefit whether or not I manage to accomplish that this year.

Thanks in advance for any advice.