Why this game ended in a draw?

White ran out of time. Ordinarily, this means that they would lose. However, in this case, you are unable to checkmate them, regardless of how many moves they give you. For that reason you do not receive the win.

#2 is correct: a lone king cannot checkmate the opponent, so the timeout which normally rules in Black's favor (because White timed out) this time is ruled a draw.

white lost on time, but black has no way to win the game

as a result, it is a draw

When you run out of time on Lichess but there is not sufficient material for any side to win, the game ends in a draw. In this case, there was not enough material for white to win because a rook-pawn and a king vs a lone king is a draw. It would have been possible for black to force a draw if the game continued.

#7 There is enough material for White to win, but White's time elapsed.

#8 but it's a rook-pawn so black could force a draw, so that's why it was a draw? Either that or I'm dumb.

#9 As thrice explained above, it's a draw because only White is capable of checkmate and White's time elapsed. Black cannot force a draw.