Here's my understanding: a piece's value is dynamic, and it depends on the type of piece and its position on the board. Positioning leads to all other aspects you listed: controlled squares, to-be controlled squares, tactical threats, and so on... (hence the popularity of "positional" chess where there's no concrete threat or plan behind a move, only improving the position of your piece, which may create opportunities later). I believe this is also how Stockfish chess engine evaluates a position, and it has higher elo rating than any human player.

Having all those said, the type of piece is still the bigger factor in deciding the value of the piece. You don't often see a queen-knight trade for example. Queen sac is definitely something you don't see every game.