Why are there five extra turns in tournament Magic?
The imbalance is intentional. It gives every player an incentive to have the time limit hit during their opponent's turn, and therefore to finish their own turn quickly. As a result, it helps each tournament round stay within the time limit and reduces the need for turn extensions.
From the Tournament rules (pdf)

2.5 End-of-Match Procedure
If the match time limit is reached before a winner is determined, the player whose turn it is finishes their turn and five additional turns are played in total. This usually means that one player takes three turns and the other two, but a player taking extra turns may affect this. If a player has already passed priority in their end step when the time limit is reached, that is considered to be in their opponent’s next turn.

When the time limit hits, the currently active player finishes their turn and, as you correctly note, is disadvantaged by getting only two extra turns vs. the opponent's three. That means you do NOT want to be the active player when the time limit hits. The best way to not be the active player is to finish your turn, so you have an incentive to finish your turn as quickly as possible and pass that &quot;hot potato&quot; to your opponent.
Additionally, with an odd number turn extension, the non-active player also gets the last turn. In a damage race situation with creatures on both sides, the player who has the last turn does not have to plan for the opponent's backswing, and can go all out with the last attack.
By providing each player an individual incentive to play quickly (in addition to regular slow play infraction penalties), the whole tournament benefits. If the turn extension was an even number, no player would have this particular incentive to play quickly, because it wouldn't matter who the active player is at the time limit. The active player would also get the last turn, so it would be an overall benefit to the currently active player and therefore an incentive to play slowly.