I just found this discussion, while looking for some answers to the same question. But in my case, I would never give up playing piano, but I am almost giving up on trying to be better. I am an adult hobbyist who played piano as a kid. I picked it up again 3 years ago.
What you write sounds just like me...poor learning as a kid and now 3 years of more serious practicing and lessons. I also have wondered the same things many times.
Why is it so difficult for me to learn the pieces? Why does my playing sound so bad?
Objectively it probably does not sound quite as bad as I feel. I am too much of a perfectionist, I always focus on the things that are not right.
But it's not all that, since I can easily hear the difference between my playing 1 year ago and now. It used to sound much worse. My progress tends to not be gradual, but happens in sudden steps. So it often seems like nothing happens and then one day I just realize things are much better. This spring I was told by a visiting teacher that I have good touch and nice finger technique...something I never imagined to hear...Maybe I could not play my pieces through properly, but at least I did something right...
I have tried to separate the two issues I have:
The sound quality in the playing
The inability to play correctly/consistently
The first one has improved more than the second one, so the second one is my main focus at the moment.
I have all the same issues the person describes - can't interpret music, can't play pieces at the proper speed, lack dynamic ranges, make too many mistakes, make too many ugly sounds, lack of balance, the list goes on. My teacher is utterly frustrated with me. I also had the same problems when I was a kid, I was declared that I have no musical talent and shall not pursue music by my teacher back then (who was a composer and vocalist, not a pianist, so I wanted to believe that was the reason).
I want to believe it's because the pieces my teacher wants me to learn are just over my head. I spoke with other teachers and a friend who is a pianist, but they seem to think I am working on the right repertoires and they also agree I have a long way to go to play these pieces well.
I do know I don't sound good at all. I have an ability to tell the difference between a great pianist and not so great pianist. It's not like I can't hear myself. But I just can't make my body to work to coordinate properly to produce right touch, press pedal at the right level, use my arms, etc. I get exhausted by the end of the piece, if I'd play at the speed I am supposed to play at. I thought over time, I build my muscle, but now I wonder if I'd see any change after 10 years and 15 years.
My teacher teaches those Conservatory students and he is not quite sure why there's such a difference of playing between his other students and me. Why others can start playing beautifully very quickly, and why I keep playing so ugly lessons after lessons. My hands are not big, but I can still grab octaves. I may be practicing a lot less, because I have a job, but I do practice hours every weekend. I try to practice hands apart, no pedal, listen to good recordings, read analysis, all sorts of things he tells me to do.
I am finally starting to admit that there's something about in-born talent / gift - and I just don't have it. I am sure the person who wrote the original post "can" interpret music, if we were to analyze music scores and talk about composer's intentions, etc. But actually playing is a different story.
I will never stop playing piano, and I will never dream of playing in front of anyone. It's more about giving up and accepting that this is as much as I can get.
I wonder if people agree that certain students are just not going to go any further beyond a certain level. Or is there a way those students can somehow break out of their limit?
While I agree that it's much easier for those who have the natural aptitude for the exact things that are important in playing the piano, I think it's still worth to keep trying.
It is possible though, that your teacher and you have not been able to isolate the things that should be worked or the methods used just are not suitable for you.
To make the playing sound better overall, you need to have the basic technique right. Even if my teacher has patiently gone over the basics of playing with me, it took me over 2 years to start developing a good touch, due to the numerous problems with my physique. It was never about not knowing or hearing what is wrong, simply about how to learn to actually do it better and get my neglected and overly tensed upper body to function better. If your teacher is not persistent enough or not used to train someone who has weak hand, tensions and trouble relating to the piano keys and using the right muscle groups, that might be part of the problem. Sometimes one has to start with exaggerated movements, which sounds bad and gradually finetune and diminish the movements to what is really needed. So for a while in the learning process you might sound worse than before, even though you are actually progressing.
To learn to play consistently and with less mistakes takes a lot of experimenting with learning, concentration and memory techniques. You need to figure out what works for you. We might never be as good and consistent as many others, but we should be able to learn to play though occasional mistakes and develope more feeling of certainty in playing. Playing easier pieces only helps if they really interest you, if you find them boring it's useless (at least for me).
At this point taking on shorter (but still interesting) pieces works better for me than trying to learn longer works that take months just to get in to the hands. Like you I also work and tend to overpractice on weekends. But that won't make me learn the pieces any faster. I try hard to concentrate less on how much work there is ahead and just focus on the pieces/passages at hand. It's much easier now when there are no lessons during the summer. I clearly try too much during lesson year to always get something ready for the next lesson, not allowing myself enough time. Trying to learn too fast often leads to inconsistency and poor quality.
If you feel your teacher is frustrated with you, you might also consider the possibility that a fresh start with a new teacher more used to teaching beginners might help.
Don't give up trying to be better! In the end 3 years is really not that long. Some people develope skills much slower than others and piano playing is a very complicated skill. Maybe for you the big break is going to be at 4 years

Edit:
Just read your post about your piano...I also have had issues with mine and I think that could contribute to the quality of playing as well. It's hard not to get tensions, when you practice daily on something that is not responsive enough.