Why would JD Salinger choose an allusion to David Copperfield to open his novel with?

Actually, the narator is Holden. In reference to Copperfield, Holden is noting that he won't be looking for our sympathy..... and he isn't going to share his life.... he's not going to tell his life story. Holden is simply going to start with when the trouble began.... the Christmas before.

“If you really want to hear about it, the first thing you'll probably want to know is where I was born, and what my lousy childhood was like, and how my parents were occupied and all before they had me, and all that David Copperfield kind of crap, but I don't feel like going into it, if you want to know the truth.