I agree that seed will certainly work on that slope. You can use the seed mats but they are expensive and then you don't get to choose your grass seed type generally. One trick I used when I had the exact same scenario, paved area, then steep slope being seeded was to run a single row of sod lengthwise along the deck. So that's like 18" wide. That will catch a decent amount of the water on regular rains and reduce the erosion. But I've seeded many areas on our hills and here's how I handle it. Place seed, lay straw generously. The grass seed will make it's way through. No clumps but like 3" thick when spreading. It will compact when watered. Then water every day until fully germinated, twice a day is even better. You only need to get it fully wet, not drenched. It's only seed at that point, not roots. Then as erosion happens, deal with it swiftly, back filling with dirt, tamping down, reseeding and laying straw.
With average soil, you will have good grass (but not great JimJim quality grass for the long term), but it will need regular fertilizer to counteract the poor soil. You may also want to overseed after one and then two years in the early fall. Or, put in 12" of topsoil and never have a worry! We probably seeded 10,000 SF after our pool was built. And that was after seeding a bunch for our basketball court and also after Leyland Cypress removal and last fall a flower bed removal. It all looks quite good now. Weeds will come up, but wait at least a month or more after you start mowing the new grass to spray weed killer. And be very careful if its over 90, wait versus risking killing the new turf. For new seeding now, I'd hand pull weeds and not spray until mid-September. With your luck, you'd spray and then the temp would go to 100 two days later.
On fences, I also have 100's of feet. I spray the grass to kill it, then put a mulch strip on each side to run the mower/tractor along. Occasionally trim it to keep it neat and spray veg. killer as needed until I re-mulch three years later. If I had a less formal setting like a ranch, it would just be veg. killer only.