I found out that DS was gifted after I enrolled him for group piano lessons at 4. He is very musical and has had music around him all his life. He was in a group piano class where it became obvious that his abilities were standing out. They had to go at a slower pace and repeat more often for all the kids to catch up while he got a concepyt on the first attempt and memorized a song after playing it once or twice. I stayed on for a year in that group because of the group dynamics, excellent teacher and the "fun" aspect of it. But it was too slow and I reluctantly switched to private lessons when he was 5. What a difference it has been - moving at his own pace is so liberating for him and he is learning at an accelerated pace. He enjoys his lessons a lot and the opposite of what we feared has happened - he sits still with his teacher for 45 minutes and focuses through the entire time. He also retains all of it!
As the OP said, my child has perfectionism meltdowns when practicing songs for the first time. We are still working on it, but I have been able to convince him to stop hitting his head when he does not get a note correct on the first attempt. It has been very hard to convince him to slow down when learning new pieces and play only one hand at a time for tricky segments to get them right. He expects things to come easily and he is baffled when he has to try so hard at anything. I am not sure if he will continue piano or switch instruments, but he will stay with piano as long as he enjoys it.
It is a good learning experience for our whole family. And no, we never mention his giftedness to anyone - some teachers figure it out themselves when they interact with him. Some other teachers (like in his PS) think that he is not gifted because he gets bored easily and tunes out.