Frederick I (HRE) doesn't die during the Third Crusade?

Frederick Barbarossa, perhaps one of the greatest emperors in the history of the Holy Roman Empire, famously ended the Third Crusade with his final breath. That is to say, he drowned in a river before reaching the Holy Land, causing most of his 15,000+ army to scatter and head back home and leaving the 10,000+ armies of the French, English and Hungarians (the former two having been fighting each other beforehand) with the job of defeating the Ayyubids (which they didn't).

How might the Third Crusade have concluded had Freddy got himself across that river?
 
Seeing as he had already beaten the Danishmends and taken Konya, and had a bigger army than the either one of the other two major participants in the crusade, if Frederick successfully made it to the Holy Land, the situation would be very different. As Emperor, he outranked Richard and Philip, so he could have taken charge of the crusade and kept both French and English armies together. I would say that the chances of re-taking Jerusalem would be alot better, not certain but much, much better. We might see a different king of Jerusalem rather than Guy. Guy's succession was illegitimate, breaking Baldwin's will, Conrad of Montferat would get a better chance of becoming king, seeing as he was related to Frederick and had a good military record, as well as being fairly popular with the locals, he might not get killed by assassins in TTL. In general terms, the land route through Asia Minor would not be regarded as impassable and thus the Crusaders would have less incentive to buy passage by boat, the Italian merchant republics would be weakened, not by a whole lot, but some, they probably wouldnt get the same dominant position in the KoJ as OTL, seeing as the king would have more than the coast. The 4th crusade might not hit Constantinople, but someone from the West will and probably while on crusade, the Byzantines have a bad history with Crusaders, seen as if not traitors, definitely not good neighbors, Frederick had to occupy several cities on his way through and the emperor Isaac was allied to Saladin.
 
One thing: at that time Sybilla, who was still the legitimate heiress of Jerusalem was still alive so Guy, as her husband, had a pretty good claim to the throne reinforced to the fact who Isabella, Sybilla's half-sister was still married to her first husband, Humphrey of Toron, who had recognized Guy as King of Jerusalem (and that wedding was annulled only after the death of Sybilla and her daughters). So for remove Guy from the throne of Jerusalem you need either to kill Sybilla and her daughters or persuade her to divorce from Guy and the second one is very difficult considering who Sybilla was much in love with Guy (at the point who tricked the council of Jerusalem and put her throne in danger for him) and Guy was a subject of Richard of England who was also the only Crusade leader to be a relative of Sybilla and Isabella. If you kill off Guy or make Sybilla (who was still the Queen of Jerusalem until her death) divorce him, her thrird husband will be the next King (and that husband for sure will not be Conrad of Montferat, because Sybilla's first husband and father of her late son was Conrad's brother). Guy's succession had not breaked Baldwin's will and was not illegittimate (mostly as I said before the barons of Jerusalem called to choice the next Queen after the death of Baldwin IV and V (the brother of Sybilla and her son from her first husband) choised Sybilla under the condiction who she left Guy. She accepted under condition to be able to choice her next husband and promptly remarried Guy and crown him as King). The barons were rather unhappy but had not another candidate to put against Guy (as I said the husband of Sybilla half-sister instead of challenging the rights of Guy to the crown recognized him as King) and in the end all accepted the situation
 
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