I've eaten at several WDW restaurants solo, including Rose and Crown, Sci-Fi, and Le Cellier. At each place I had a great experience, didn't feel awkward or ignored, and enjoyed the food and atmosphere. Of course, I might just be used to solo dining since I've done a lot of solo travel or solo dining at home.
The only restaurants I'm not crazy about doing solo are buffets. First, there's nobody to watch my purse when I go back to the buffet line to get food. Ladies, you know what I mean! So I feel like I have to pack up all my belongings just to go get a slice of pie. It's annoying.
Second, I've had servers clear my table and try to seat someone else in my place when I returned to the buffet. They also thought I had run out on the check. This didn't happen at WDW, but it's happened a couple of times at buffet restaurants at home. They think because I'm gone and my purse is gone that I'm finished and that I left for good. Aaarrrggghhh! So I tend to avoid buffet restaurants when I'm alone.
I'm kinda iffy on family-style restaurants when I'm solo, too. Sometimes I just don't want to share a table with strangers. Then sometimes I don't mind so much.