You were on track until you said - also prevent erosion. If the slope is steep and the soil is anything but hard packed clay (which won't grow anything) the only way to prevent erosion is with deep rooted plants. Most ground covers have shallow roots. Even turf grasses have roots that might extend down 20 inches but only during the growing season, the roots retract to about 6 inches during the dormant season. A lot of common garden plants (including shrubs and small trees) survive with only a matt of fine roots up near the surface. We like to look at carpets of plants covering the ground and THINK that they are protecting it from excess water but that isn't how it works. With enough water, erosion can undercut any matt of ground cover.

I would cover the ground with gravel and let sedums take over. Though they look like full sun plants but in truth plenty of them enjoy the shade.

have you ever considered creeping phlox?? It blooms with many different colors, some of them rather new, like appleblossom and a fairly true purple. Mine stay green year round, and in the spring are covered in bloom, even in dappled sunlight. And yes, I know all about hard pan
clay....I just chisel out a spot for the new creeping phlox, fill it with compost, and plant it.....with continued mulching of the hard pan clay, it is slowly making an inch or so of top soil out there after 5 years of working with it.
Good luck.....

variegated vinca major..(hey..you said 'invasive'). I have it in just that spot down a steep shady hillside. Depending upon what trees are doing the shading, you'll need to water it regularly until next spring when the pretty purple blossoms come. You can also add in variegated spotted dead nettle. The variegated gives some brightness to the plantings in the shade.
Unlike ivys, the vinca and the dead nettle don't climb the trees.

You can also start patches of hostas and hellebores in your groundcover. The hostas can be divided pretty frequently and hellebores have a way of getting you to buy more of them every year.

What ever you do don't be tempted to try baltic ivy unless you are committed to CONSTANT VIGILANCE (and trimming, pulling, etc.) to keep it out of trees, lawn and off the house. It just likes zone 7 conditions way too well.

There is a plant called Wire Vine (Muhlenbrekia or something similar) that is often sold as a filler for combo pots or hanging baskets. I got some years ago and treated it like a tender tropical for years until I noticed a piece of it surviving winter and actually spreading and looking much better with neglect. Now I leave it outside and let it go to town. It looks good up until the temps get down around 10degrees and then the leaves will die back but the stems re-sprout once the temps get back to normal winter temps.

It isn't the classiest plant for the garden but it does seem to choke out most weeds and it likes shade and poor soil - even dry soil (but you would have to water it the first year). It doesn't really make much of a vine, just simple looped stems about a foot long with little round leaves every inch or so. The loops stick up about a foot off the ground which makes the area hard to walk through, so it isn't a ground cover for high traffic areas but otherwise it is worthwhile plant for dry shady slopes.

If the area is dark shade the only thing I have ever seen survive those places is vinca, with Vinca minor covering the ground better than Vinca major.

Autumn Ferns...they're evergreen, spread slowly, and are beautiful in mass. They take shade and a little sun and are very low maintenance...not even too picky about their soil from what I've experienced, though they perform much better with lots of good organics. There are other ferns as well, but it's tough to beat the autumn fern. Liriope (monkey grass) is another option that hasn't been mentioned, perhaps because it's so common and folks think of it truly as "border grass" when in fact it works great in large masses. And if done correctly with some thought, you can combine several different groundcovers...one of my personal favorite combos is liriope, autumn ferns, and hosta iwth a few punches of color around the edges. The ferns provide good color as the new growth unfurls in the spring, the hostas give summer color and the liriope blooms in late summer...and this is all on top of the great foliage and textures you get wtih this combo...but the hostas will die back in the winter. And this grouping is specifically for a shade garden.

I have a hillside of vinca minor that i'd like to get rid of- if you want to drive up this way you're welcome to it. It's a bit too invasive for my taste, esp given that i back up natural woodland i don't want it taking over. It really does do well in the conditions you describe, though. I agree that a combo of things may be the best way to go- eventually one will probably take over, and then you know it's very well suited for that area. One of the mentioned plants- hellebore, wouldn't really be walkable through, but if you spaced out the adult plants, you'd have a solid colony in under 5 years. They reseed and fill in nicely, are evergreen and about impossible to kill unless you try too move them. Their roots are sturdy and relatively deep, too.

Why invasive? You'll regret it.

Try our native allegheny spurge...pachysandra procumbens. Interesting evergreen foliage, fragrant white or pink flowers, and fairly low maintenance. And it's well behaved! The only drawback is that it can be tough to find in the retail nursery trade...even Niche Gardens is sold out, apparently...

Here is a link that might be useful: Native Pachysandra

The reason you don't see Allegheny Spurge offered more often is because it is so slow growing. Most nurseries need to sell plants that fill a pot quickly. I planted some in my dappled shade front yard (sandy loam soil, lots of compost) and after nine years it has barely doubled in size. Plant Delights has sold it in the past. I doubt it is in the current catalog but they probably have some for sale. You'd have to visit them on one of their open house days or call them to find out. Most people just plant the Asian version which fills in faster but neither of them could be called invasive.