Alabama?

Yes, just north of Birmingham.

Just watched a video on YouTube from Southern Living and they say the Bronze Beauty grows 8-10 feet, LeAnn 10-15 feet, and Bigfoot 15-25 (?). We didn't want anything THAT tall, so would Bronze Beauty work behind some Edward Goucher abelias?

Once planted we won't be able to get up this slope to trim/prune, so want something that can be left alone.

If you're talking about Ternstroemia gymnanthera (Japanese ternstroemia), I think that would be a good choice. You may have to do some supplemental watering if they're in full sun.

Another choice would be compact, upright hollies such as Blue Maid, Spartan, or Castle Wall/Spire.

Sounds like you don't want anything much over 10 feet?

Well...we just had to remove 4-5 GIANT elaeagnus from that spot because they tipped over from the heavy rains we had in January. When we moved here, we had no idea how big and unwieldy they would grow, just wanting something quick to fill up this slope and give us privacy. Obviously the neighbors hated them because they would throw cut branches into our yard. (And when they tipped over, there was a big pile of cut branches on top!)

What's a little funny, though, is that a landscaper who came over to advise found the stake for the property line and we own about 2 feet more of our neighbor's yard than we--and probably they--realized. So they actually had no right to object.

We went to a nursery a few days ago and the owner recommended "Cleyera Japonica" saying that he has not noticed any differences between it and the cultivar Bronze Beauty. He said it will make an "impenetrable" screen for us but not as crazy as elaeagnus.

Yes, Elaeagnus grow big - but yours may have been root-bound. Inspect roots and pull them open when planting container-grown shrubs. I had some Ternstroemia go over in snow after many years as foundation shrubs. When I dug them up and shook the soil off the bases of them looked like rutabagas, most of the main roots having been wound around into a fist by tiny pots during the early days in the production facility.

Anything you buy that turns out to have tight root knots at the bases of the stems should be taken back to the store. I don't know if any given vendor will provide satisfaction once the plant has been "ruined" by efforts to correct bad roots at planting time but the system is never going to stop putting out plants with deformed roots if nobody ever complains.