Disney is a great place for an anniversary celebration!
DH and I went in 2014...he actually proposed at CRT, so I biasedly think it is romantic. I probably would have thought it was more cheesy/awkward without the proposal though...the princesses were cool to meet but also did take away from the romance. I don't usually wish changes on classic Disney spots, but it would be nice if they had and "adults only" section that you could opt to skip the princess interaction and just enjoy the atmosphere and food...then I would DEFINITELY recommend it!
We also LOVED BOG for dinner last trip. If you can make an ADR, then go about 30 minutes before your reserved time and request to eat in the West Wing it is so awesome! It is a darker and more intimate room, and scares some kids (we were told by our server), so more adults. Belle's rose is on display and there are lots of cool details to take in while you are dining. As far as I know, it is also the only restaurant in MK that serves alcohol (though I could be wrong about that)...they have several wine selections. The food itself was phenomenal and the service was excellent. That's also where you can try "The Grey Stuff" and if you're wearing a celebration button I bet they would bring you a complimentary serving....we got some because of our "Just Engaged" button.
We are heading back in Oct. for our first wedding anniversary. Because of
MNSSHP we can't go to either of those two restaurants, so our TA booked us a late dinner at California Grill on our actual anniversary. You can have a reservation there at any time and still have access to the exclusive observation deck for fireworks. So if you wanted to go to a 5:30 dinner and then leave and come back later, you can do that.
Another suggestion for finding the romance at Disney....we booked a carriage ride at POR before our California Grill dinner. Not sure if that is something you might want to check out, also