Don't try to do everything at once. Install the top first, adjacent to your fence, babying it along with TLC until it is up and running (thick with no weeds.) Kill any weeds first before planting. You can remove their tops, but don't dig or pull. Leave their dead roots in place. Make sure they stay killed during the establishment period of the groundcover. Also, keep at least a clear 3' band of DMZ between installed plants and weeds, so that you can respray any returning weeds without hitting the groundcover. Asiatic Jasmine will probably work well for you. I can't say what will be BEST because I'm not there or familiar with all of your conditions. Look around and see what others are using in similar situations. A goal would be to get groundcover established where it is easiest and then encourage it to grow into the more difficult areas. Part of this strategy will be focusing on the easier areas, one at a time. Where you need to use erosion control blanket, excelsior, though more expensive, will be much longer lasting product. Jute has no lasting power. In two months on the ground, burlap is falling apart.

Realize up front that there will be a limit of how far you can extend your control of weeds. Decide how far you're willing to go. If you end up with a nice 10' or 15' band around the house that is weed free and all beyond that go to every kind of greenery, the collective sum of which is just background, that should be fine.

@Yardvaark thank you for such a thorough response! Just a couple of more questions: 1) where along the slope do you think we should put the excelsior? 2) As you mentioned, there is only so much we can do to control the growth in a wooded area. What would you think of adding something taller right behind the fence that would visually (and physically) block some of the weedy growth on the upper slope? Something like a winter jasmine shrub behind each section of fence? Just focusing on the top section of slope? Possibly adding ground cover beyond the winter jasmine? Thanks again!

Put excelsior where it is steep enough to be NEEDED. It's not about aesthetics. It's about function.

I would not put something taller behind the fence. I'd maintain a refined strip of landscape (groundcover) to some distance, a few feet ... it could be 4' or 10' or whatever you determine, beyond the fence and let nature commence beyond. Everything will be green and look as good as any roadside greenery. The open feeling would be better than being penned in by a hedge of some sort, which would also bring additional maintenance ... trimming and keeping weeds and vines out of it. They would be more prone to getting a foothold in a hedge since you can't see when start.

@Yardvaark Thanks so much for your replies! This may sound dumb, but how do I know if the excelsior is needed? The slope is steep, but the house has been here 25 years and I don't think there is an active erosion issue. It's mainly been an aesthetic/maintenance issue but we certainly want to think about erosion prevention given the proximity to our house. Do you think the groundcover would be sufficient to achieve our goals?

Also, FWIW, the view of the backyard is mainly is from above, not from eye level. The main level of the house is at street level, and then as the yard slopes, there is a basement level with two bedrooms that look out to the yard. The house has a wraparound porch that looks down over the slope into the woods. I think that's why I was wondering if a cascading shrub might be more visually appealing than just groundcover. Now that I'm looking at it, I agree that planting shrubs would create a visual wall behind the fence and make the tiny backyard feel even smaller.

An erosion control mat is needed where new plants and mulch will not stay, on account of steepness or washing, of their own accord. If you can get plants established without it, which of course you can easily do on the more level area, go ahead.