It sounds as though you understand the rules correctly; in order to go out, a player needs to hit on an already played meld/phase (their own or an opponent's). It shouldn’t require an unreasonable number of turns to accomplish this though. After laying down your phase one, you have only 4 cards remaining in hand. There are every time you draw a card, you have about a 4 in 12 chance of it being a card you can use to hit, or a 33% chance... it’s a little less because 3 of each number you need has already been played, but then there are also the wild cards to help you out. But still, you should be able to hit with the card you drew for a turn about one out of every 3-4 turns on average. This would take about 16 turns to get rid of all 4 cards. Worst case scenario, most of the required numbers are near the bottom of the deck, but they would still get drawn.
Now if you and your opponent had happened to both lay down the same numbers as each other for your phase, it would be more difficult.. there would only be 2 different numbers to hit on, and 6 of the 24 cards number you need would already be used up. But even so, the right numbers would still get drawn eventually. Turns would be very fast at this point, so you aren’t wasting a lot of time playing this out.
However, if you feel like this part of the game is someone pointless; you would be generally right... the main reason to try and go out is to end the hand before your opponent can lay down their phase. Once all players have laid down their phase, it matters a lot less who goes out first. The only reason to care at all is to address the situation where there is a tie at the end where multiple players got phase 10 in the same round. If that doesn’t happen, then going out first makes no difference at all.
So if you are playing a 2 player game, and you have both laid down your phase already, you could always just flip a coin or roll a die to determine the points for hand. After all, if you both have 4 cards in hand, then it’s just 50/50 which of you will get 0 points this hand (with a slight edge to the person who gets to draw/play first). And the person who doesn't get 0 is very likely to only get 5, or maybe 10 points against them.