I would highly, highly recommend doing a mixed planting rather than put all your bags into one egg as it weren’t. You never know what will be the best performer and a lot of times it’ll vary from one side of the yard to the other. What’s more, you’ll get to enjoy the benefits of different things if you, well, plant different things. And you’ll be protected from catastrophic failure. Three or four different species are always a better bet than one. Plus, when one plant randomly dies in year eight, it won’t look incredibly weird, unlike with single-species plantings like the typically Leyland cypress hedges you see everywhere.
Consider growing a combination of:
Yaupon holly, mentioned already, checks all the boxes and doesn’t look anything like most hollies, so that’s an option, it’s pretty quick growing too. Very tough, deer don’t bother them here and I’ve got insane deer that eat everything including poisonous stuff.
Inkberry is another holly that doesn’t look at all like a holly, and it’s likewise tough.
Carolina cherry laurel is an evergreen cherry with leaves full of cyanide so the deer learn not to touch it. Gets a bit taller than your criteria but responds well to pruning and shaping.
Wax myrtle, usually pretty much bulletproof. Should be readily available and cheap.
Privet, though it’s invasive as heck so unless it’s already common where you are then don’t introduce it
Silverberry, which will probably be Elaeagnus x ebbingii, is extremely fast growing and is deer-resistant. It will get above the 20 ft limit in good conditions but responds well to even hard pruning and shaping. Also invasive as heck.
Podocarpus macrophyllus is a bit slow growing, but it’ll check all the boxes you list, and taxonomically it’s really cool.
There are a handful of Eucalyptus that might manage. The “alpine gum” types like E. perriniana and E. pauciflora are surprisingly hardy. Best grown from seed, they don’t transplant well after the seedling stage but they’re (in)famously fast growing. Deer don’t touch them, way too leathery and oily and stanky.
Camelias are slow, really slow sometimes, but they fit the bill for what you’re asking. Deer resistant, perhaps not deer-proof.
Osmanthus would also be great if you don’t mind something a bit slower growing.
Illicium floridanum tops out at like 8-10 ft so it barely makes it into the consideration, but it’s a really cool one too. Same genus as star anis (but don’t eat this one, it’s poisonous as the deer will hopefully find out).
I personally don’t like Aucuba or Photinia, but they’re also options if you’re interested.