Loving ^ but if that won’t suit then…
Cotoneaster – as above – but although it looks okay from a distance it is woody and can look ‘untidy’ close up. Small white flowers in summer and berries in Autumn so some interest
Vinca (periwinkle) – Low growing shrub that spreads in a similar way to the cotoneaster, but is much ‘softer’. Pretty pale blue flower (also available in white or purple).
Euonymus – low growing varieties. Don’t root as they go, but do provide nice cover. Lovely variegated leaves, so attractive all year around.
Ground cover roses – Will love the rich soil from the decomposing grass cuttings, They don’t need pruning like normal roses and can be left to cover the area – but prickly if you do need access.
Pachysandra – Good for ground cover but green and boring!!
Juniperus – low growing conifers (spreading) which will make a solid mat over the area
I assume that the grass cuttings are now decomposed and just making the soil loose? If still resembling grass cuttings then they will need digging in a bit before you start (will burn new plants)
From above I wouldn’t recommend Laurel or Rhodo you will just be creating a huge deep hedge which will require yearly cutting to keep to a reasonable height (I am assuming you are not looking for an 8′ barrier here?)
There are many other options… but all will need some form of weed control help while they get established – might be worth using a weed control fabric and planting through?
EDIT – Or stick with the Ivy, and add some pretty variegated ones into the mix..?