Kind of irresponsible to leave him right then

I agree, I thought they were just getting to his growth as someone who has beginning to see the world beyond his problems and to put his life into perspective. She was aimless in her time since not being able to attend Juliard. I also saw no redeeming factors in her returning to her obviously toxic home environment. I expected them to begin to play together & have this "healing" through music montage at least. I'm disappointed in the ending. I loved his dialogue though. This kid is a wonderful actor!

Maybe you didn't notice that both the maid and mother were both returning the next day. The father had been dead since he was three. Also, the new nanny had arrived, and appeared responsible and nice. She was sent by the same agency that sent Eleanor, probably with the mother's OK.
You're right to some degree of course, it would have been better for Eleanor to have waited another day to leave, but I assume that for plot drama purposes the writer wanted Eleanor's parting from Reggie to be with them more private and isolated, leaving him alone with only the new outsider nanny elsewhere in the lonely house. Reggie is so self sufficient that he was probably no more anxious or alone than with his disconnected mother and full staff present. He may have preferred to part this way without other eyes on them.
The was I see it is that Reggie fell in love with Eleanor, but she was not in love with him. Not in the same way. She had formed a real bond, but he was only 12, and she was not comfortable being longer term staff in that household; she'd taken the two month job to fill in until the mother returned because she'd lost both job and housing, and was in a crunch, and the mother was now returning. It's a story of unrequited love, and a boy not yet grown up to match his intellectual and emotional level. Reggie is heart broken, not anxious about being left without supervision.