How to balance Dixit?
That particular rule isn't particularly balanced, and is included for convenience. However, playing with fewer cards in hand also imbalances the game, as storytellers have fewer cards to choose from to come up with a good hint, and other players have fewer options to play in.

Turns as the storyteller aren't the same value as other turns either; the storyteller is probably more likely to get their 3 points, but other players stand to gain more points by misleading other players and guessing the card correctly.

One way to fix this is, as you mentioned, to reshuffle cards until everyone has had an equal number of turns. When I play, we agree to finish at the end of however many more rounds.

Additionally, if you have extra Dixit card packs, you probably don't want to keep playing until you run out of cards.