I've eaten gluten free at Disney for several years and never had an issue. Note it on your ADR's and confirm when you get to the restaurant and they will either give you a special menu or have a chef come out to talk to you. I've had great experiences overall with one exception; Chefs de France. Chefs doesn't have a gf menu and the server had to go back and speak with the chef several times to see what could be made gf for me. Most of what was offered was that they would omit a sauce or substitute with a gf sauce, but that changed the dishes drastically. It wasn't a bad experience, just one that was not as easy or gf friendly.
My favorite places to eat for TS are:
Ohana (gf platter for breakfast and for dinner the meats are all gf. You will get a separate platter of chicken wings and a side of rice), Raglan Road (gf fish and onion rings are AMAZING), Tusker House, Sanaa (HIGHLY recommend the bread service with gf naan and the butter chicken), Biergarten, Spice Road Table (rack of Lamb is terrific!), Bongos, Rose and Crown.
QS favs are: Sunshine Seasons, Cookes of Dublin (same fish and onion rings as Raglan Road), BOG lunch, Flame Tree. I'm excited to try Satuli for the first time on our upcoming trip.
We usually go during Food and Wine or Flower and Garden and a lot of the booths have gf options which are clearly labeled.
I am not a fan of Erin McKenna's baked goods. I also pass on any Enjoy Life cookies if I'm offered them as a dessert option. OMG it's Gluten Free brownies are my favorite snack that you can find at some QS locations, like Sunshine Seasons. The gf beignets at POFQ are good, but I found them a little oily. It didn't stop me from eating a bunch though! It's rare to get a fried gf dessert!
Trust me...you will eat better at Disney than anywhere in your local city!
GF NAAN!?! That's a game changer.

I may have to revisit my ADRS.
They will even share the recipe with you if you ask. I emailed guest services requesting it and they sent it to me along with the recipe for red chili sambal, which is my favorite of the bread service sauces!