Disney has taken the approach of many restaurants these days -- the noisier they are, the more party-like the ambience and the more likely patrons are to spend their money. Hence, nearly every restaurant has an open kitchen. There are very few that play any soft music that would help tone down the din.
Noise is one of dh's and my biggest complaints about Disney restaurants.
You can get some quieter experiences at signature restaurants but it's not easy. It's even harder at the 1TS restaurants. In fact, it might just be nearly hopeless.
Here are my thoughts given you are not on the dining plan but want to keep the price down. (Even though you aren't on the dining plan, I refer to restaurants as 1TS that are less expensive than the signature restaurants.)
1. Dine at off times (later in the afternoon for lunch and after 9 pm for dinner) and, at signature restaurants, ask for the wine room if there is one.
This is no guarantee mind you. We were once dining late in the wine room at Jiko. It was 10 pm. We got concerned when we saw them setting up a large table next to ours with three toddler seats and a high chair. We said something to the waiter and they then rearranged things and set up a table across the room for the party. About 10:15 pm here comes a large group with a baby and three toddlers. Apparently, the wine room waiter was the one working latest so that's why they were put in the wine room. Before we knew it the toddlers were squealing while having sommersault races across the room. It was only when they started hanging from the beaded curtains that the manager stepped in. So signature dining in the wine room is no guarantee of quiet, but it sometimes helps.
2. I agree that afternoon tea at the GF is generally quiet. Music does start at 3 pm. We enjoy the pianist but not the singer who sometimes sings with him.
3. I do not agree that GF Cafe is quiet. It used to be and was a favorite of ours. Not anymore. As with all 1TS restaurants these days, it's packed and noisy.
4. We have had somewhat quieter meals at Olivia's and Turf Club. Sanaa is only quieter if you dine very late -- otherwise it's full of big families who are occupying the DVC villas.
5. I find Mama Melrose to be sensory overload. Very busy with Frank Sinatra playing. Waiting for a table in the front is a particular trial.
6. Liberty Tree Tavern might be okay for a late lunch. Don't do dinner.
7. Fulton's Crab House is often quieter for a late lunch, as is Portabello Yacht Club. Both are at DownTown Disney.
8. We've had a quiet lunch at the Chinese restaurant at Epcot. Forget dinner there.
9. The upstairs at Yak and Yeti can be quieter for a late lunch or a late dinner.
Unfortunately, as I think about it, quiet dining at WDW is pretty much hopeless, especially since the advent of the Dining Plan. There is some possibility at the signature restaurants if you dine later. Otherwise, you just have to be prepared for noise.
)...we were very pleased that they left before we were served our meal...
...so unfortunately you just never know.
I'll tell my DH that! Disdain for crowds might trump his thrifty genes! 
