Probably for a number of reasons:
1) Non-paying members have limited access to the Chess.com database
2) The Chess.com database is not very extensive
3) Correspondence databases are an excellent resource for correspondence chess, and are not accessible through Chess.com
4) Every other site has this feature too.
I believe that there is no or little legitimate use for both FEN and PGN while an online chess game is running. Engines are not allowed, so the only thing you could paste it into legally is an own opening database, but why not use the one provided by chess.com?
Yes, you could still cheat by entering moves/positions manually into an engine. But FEN/PGN make that easier than necessary.