Would be helpful to see pictures of the house for context. Also helpful to know the direction you are facing so we know whether the area gets morning or afternoon sun.

I totally do not get why you'd be planting shrubs at the perimeter of this bed. Shrubs might be like "islands" around the trees, But the edge area should be like groundcover.

You could also return the sunny portions of these beds to lawn - another type of groundcover.

What are the existing small plants in a row along the edge of the bed?

The best sun groundcover is grass. There are a number of good ones for shade. The rocks are just going to be a PITA, so get rid of them. Whether or not you want/need shrubs under the trees has yet to be determined. If the existing shrubs are for sun, they can be moved to the south side of the trees, quite close to the trunks.

Definitely agree that rocks are a PITA. Been there, done that.

Great suggestions, thank you! How would you go about establishing ground cover over such a large area? Full coverage would probably take quite a few years?

Me, too ... rocks PITA.

One tremendous advantage to a longstanding mulch bed it that it is weed free so it is ready to plant now. One doesn't have to spray weeds, wait for them to die and also fear their return.

The first thing to consider is picking the right groundcover. It must be one that is tough, moderately aggressive but hopefully not overly so, low height -- I would say here 6" to 12" is good -- and accepts the light conditions. Some groundcovers will grow in sun and shade. You don't want a groundcover that stays in a slowly enlarging clump because it would take forever for it to cover.

There's probably a wide range of climate zone in NY. You must get groundcover that works well in your area. A great way to see what's possible is to visit a nearby botanical garden, or look around where you live and see what people are growing on slopes and in shady areas that fulfills the plant requirements. Vinca minor or Liriope spicata might be a possibility. Euonymus fortunei 'Coloratus' might be another and surely there are more. Be sure to check what grows well there, though, because you don't want to start down the wrong road with a plant that will be no end of trouble.

Considering that acquiring enough plants to cover this area could be costly, you might consider learning to propagate whatever groundcover you use. Then, the best source of plant material will be anyone you can find who's growing it and needs it trimmed or thinned (if it must be dug.) I had no trouble at all acquiring groundcover in this manner. If you get cuttings that must be rooted, it's typically a 3-4 week process, but can net you hundreds or thousands of rooted cuttings in a short period of time.

Another factor will be that after you plant the area, new plants will need to be watered FAITHFULLY. Too, extra water will help them grow really fast. You don't want to plant a bunch of little plants that end up dying so make sure there's a watering plan in effect before you plant.

Another question, the bed is covered in natural wood chips right now. Similar to this:

You would not remove it because you'd still need a mulch layer in the end. You'd just scoot enough away to create a little planting hole and then scoot it back a little after the plant is installed.

Plant a lot of plugs of the ground cover. This is the beauty of natural wood mulch. By the time the wood breaks down into compost, the groundcover will cover over the spaces in between. Win win.

As for how to plant, here's another way to look at it. In a natural forest, trees for the "overstory." They are going to provide a lot of shade, and it looks to me like your trees are kind of close together and are competing for the sun that hits the overstory. You may want to take a look at them and where there are two big trees close together, make a choice and cut one down. You don't have to do this but it will make more light come in to the ground so there might be enough for some shrubs to catch some light under the trees. They provide blooms if you pick the right ones, usually in the spring before the trees totally leaf out. Then on the ground, if you haven't spaced your shrubs too close, there might be some light and space left for some shade flowering plants. That's how you get some interest in the "understory," create some places where sunlight can filter down. Other than that, groundcover or just toss it in and make the bed smaller. Usually someone makes a bed that big because they want to put a woodland garden there, like I have described.