Goats vs. Groundcover

I’ll start by saying I know this is probably a fools errand, but why not give it a shot. Zone 9B for reference with the hill in question being in partial shade. (viewable in the back on the photo I posted).

I have two extremely spoiled goats and one dirt slope that I am in a constant battle with over weeds. My goats have zero interest in being helpful citizens and actually eating weeds, but seem to take extreme pleasure in ripping up any plants that I like. They mostly keep my citrus trees in check, goats usually eat up and are not necessarily great ground grazers, although mine do love to rip things up and spit them out.

I would love to get some low ground cover to help with weeds and erosion prevention. I was thinking of trying out Creeping Thyme (Thymus serpyllum). I have read that it is deer resistant due to the smell, but is not poisonous. I wouldn’t trust my goats with anything toxic to them, they share a single brain cell and have exactly zero survival sense. Does anyone have any thoughts or recommendations on things that may do well for ground cover and is a fast grower? My hope would be that it is not palatable enough for them to take a large interest in while it gets established and that stays low to the ground that can survive a dry climate.