I visit Gatlinburg about 5-6 times per year and probably go trail riding maybe once a year down there. I always do a little research before selecting a place but I have yet to find an establishment that isn’t going to throw you in a group with a bunch of beginners and only allow you to walk.
I can tell you a bit about the places I’ve visited even though I’m not sure any would exactly suit what you’re looking for… I would probably recommend either Sugarland Stables or Cades Cove Riding Stables simply for the professionalism of the staff and care of the horses. It’s been a while since I’ve been to either but both seemed to be quality.
If you aren’t going during a super high volume time, your odds of being in a smaller group would increase… If most of the school spring breaks are over by the time you go, you may be in luck!
Sugarland Stables - I’ve been here the most times although it’s been five years since I’ve visited. I think the trails are good and the stable itself is very accessible and close to town. You get to walk through some water on the trails, if I recall you can see some good views depending on the length of the trail you select. The horses always seemed to be well cared for and the staff was friendly.
Smoky Mountain Riding Stables - Would not recommend, went once and never went again. The staff were terrible and rude.
Cades Cove Riding Stables - Overall a good place, mostly flat trails in wooded areas, horses seemed well cared for. I’d love to ride in the fields at Cades Cove but this wasn’t that, sadly.
Jayell Ranch - I went here recently because it had good reviews and it was nearby–we were spending the day in Pigeon Forge. The trails were really nothing to write home about but I did think the horses seemed happy and the people working with the horses seemed to know what they were doing. I did like that you ride in sort of an open field at one point. They offer ATV rides at this place so there was some noise from those. The horses did not mind but if you want to only listen to the peaceful sounds of nature, this may not be for you.
This post may be entirely unhelpful since none of these places really fit your criteria. I’ll be interested to read if you get other responses–I’m always up for trying a new trail riding place, but there are SO many options, it’s hard to know which ones to try out!