The reason Jerusalem was not taken was because the Crusaders did not have enough troops. While the Europeans were willing to risk the losses in taking it, the native Crusaders knew that Jerusalem was indefensible without also getting the castles and towns past the Jordan. Richard also needed a quick resolution as the King of France was causing trouble back home, and Richard couldn't tarry any longer in the Outremer.
However, the Third Crusade has all sorts of things, that if they had gone differently, could have changed the outcome of the war.
1) If Barbarossa had lived, you not only have one of the greatest kings of Europe helping lead the Crusade, you also have a lot more soldiers.
2) If the siege of Acre had gone better, Crusaders could have taken Acre earlier and saved the lives out of a lot of Crusaders who died during the siege.
3) If Richard didn't have to rush back to France, his leadership might have have saved more of the situation. Perhaps the King of France dies while in the Holy Land, so Richard doesn't have to worry. Or Richard never becomes King of England, and he travels there just as Duke of Aquitaine, Or the presence of Barbarossa serves to pacify the King of France so he continues to stay there since he doesn't need to feel to be in an inferior position to Richard, but is comfortable with Barbarossa being the senior figurehead and both Richard & Philip as equal seconds.
4) If William II of Sicily lives and lends his navy and forces to the Crusaders.
5) If Barbarossa lives, then maybe both Armenia and Antioch contribute to the Crusade as they are more confidant of success.
6) Saladin barely lived through the Crusade, dying soon afterwards. He had run into constant problems with keeping his own troops in the field. A more significant defeat at some point, might not only cause him to lose most of his army, he might die of a broken heart.
If all of these happen, and you only need two PODs - Barbarossa living and William II living - then the Third Crusade could easily become an overwhelming success that re-establishes the old borders. Even one of those PODs might be sufficient.
The strategic situation still remains poor, but the Crusaders had sufficient room to survive and expand as long as the Muslims around them remained divided. It was only when they were united that the Crusader strength proved insufficient.
Furthermore, although unknown to the Crusaders - but known to us - they only need to hang on for another 60 years. At that point, the Mongols arrive and totally change the strategic dynamic of the Middle East. At that point, the only hostile Muslim power is Egypt, and they are no longer caught in a vice. If the Mongols arrive and the Kingdom of Jerusalem is at the old 1187 borders, they just bought themselves another 100 years at least (some of this as nominal vassals of the Mongols, but that's OK). That's a long time to build up strength. At this point, the survival of a Christian Levant (in some form) becomes a real possibility. By the 20th Century, it might just be a super Lebanon from the coast to the highlands from Antioch to Ascalon, but it'll have strong roots.