I am notoriously picky. I do my best to never miss one wrong note, one wrong rhythm, one wrong articulation, etc. However, I also require a lot of independent work from my students. I do not spoon feed every new piece to them. Typically, I will play through a piece for them so they can get the feel, and I will mention/work through a new technique or tricky passage that may occur. The rest of the learning process is up to the student. If they are unable to learn their music this way, I have failed them as a teacher. I must make sure that they are secure enough in their knowledge of note reading, rhythm, articulation, etc. that they can learn their music mostly on their own (after all, they are their own teacher 6 days a week). My job is to keep them accountable (be picky) and to teach them new techniques/ideas as they appear in their music.
I teach this way because I know one day, they will leave me. But I do not want the last day the take a lesson to be the last day they play music. I want them to be able to continue exploring new music and enjoy playing. However, I see so many students who are being spoon fed every measure of music they learn. How will a student like that learn once they stop taking lessons? They won't. They will become the grown up who "used to play" their instrument. I want my students to leave me able to pick up a piece of music at whatever level they are and know how to work through it on their own. Some will do this, and some will not. But at least then the ball is in their court. When we teach our students to "not need us", I believe that we raise the likelihood of music being a long-lasting part of their lives.