Thanks for the question! Two part response:
1 - The rule about the last piece captured is directly connected to the checkmate move. If a player captures all of their opponent’s pawns, they make it impossible for their opponent to checkmate. Keeping one pawn on the table means that there’s always a chance.
2 - Since you can win the game by capturing all of your opponent’s pieces, your final move of the game could be made with any of pieces (as you capture your opponent’s final pawn).
Hope that makes sense!