That's actually not completely accurate. In my building teachers fight to the last day to get kids through. Even if they are not trying themselves. Recently I have had many conversations with teachers still worried about kids that have stopped attending or stopped trying all together. These people continue to fight forces that they can't possibly control because they care.
It's hard to say to a teacher, "You've done all you can, you gotta let this one go" and expect them to let it go. It's like the ER doctor on any TV show that just won't stop fighting to revive a patient. Good teachers know that there is not a finish line in education. Even if you keep a kid from failure they still have to survive the summer so you can get them through one more time. I am very blessed to have teachers that fight an uphill battle and come right back and do it again every day and every year.
Guess what? It's not your fault. Many of the reasons that our kids fail is way out of the school's control but feel accountable for them. We do a lot for them but ultimately at the end of the day they go home to somebody else if they go home at all. That person or persons may not be as dedicated as you and you can't help that. Some families work hard but still can't provide for our kids. Some kids know how to work their parents and know they can't get one passed you. I know how many times you give up your lunch to work with struggling students. I know how many times you have chased them out of the building to give them the homework they are too cool to take home. Guess what? It's not your fault.
The fact that a teacher will take responsibility for a student's failure is what makes teachers so special. Teachers know what our kids face and stand tall with them and will shoulder their worlds if that's what it takes to help them be successful. I will admit that I miss the impact I had as a teacher though I try really hard to. I don't always get that special feeing with administrative work but having a dedicated and relentless staff is the next best thing.