Frederick Barbarossa was the last man who could have possibly unified the Holy Roman Empire. He began his reign by forcing the German nobles to bend the knee. With the empire unified, he invaded Poland to gain more land. Things were going great until the Pope conspired to pry Northern Italy away from him with the creation of the Lombard League in 1167. The fiercely independent Italian city-states, aided by the Pope, defeated Frederick in 1176 ending his dreams of a trans-alpine empire. Italy was now totally independent.
Finally, in 1187, Frederick received a desperate cry for help from the persecuted Christians in the middle east. Being a God-fearing man, he assembled a massive army of 15,000 soldiers. Because his army was so large, and it moved by land, it was going slowly. Frederick didn't cross the Hellespont, entering enemy territory, until Spring 1190. He fought a short battle with the Turkish Sultanate of Rum, sacking their capital and then continued towards Jerusalem. But while crossing a small river he was thrown from his horse and drown by his heavy armor. Despaired, his army turned around and marched all the way back to Germany. There his son tried to abolish the elective monarchy and tried to set up a hereditary monarchy but he was not as strong or as charismatic as his father. His reign would end in failure and the decline of the HRE into a constant state of warfare for the next 700 years between bitter, petty nobles and an impotent line of emperors who were unable to assert their authority.
What if Frederick had not met an anti-climactic death? He certainly would have turned the tide of the 3rd Crusade since Richard the Lionheart nearly took Jerusalem all by himself. Together they would destroy or weaken the Ayyubid Sultanate and ensure that a 4th Crusade would not be necessary for another 50 years. Thus, Byzantium is saved from a 4th Crusade for the immediate future.
Frederick would return to Germany, victorious, and continue to consolidate his power. For his political enemies, the quarrelsome German nobles, he would banish them to lead the newly minted Middle eastern Crusader states, seizing their German lands and giving them to his allies. He would pass the throne to his son. With the throne of the Emperor more powerful than ever, his son would be able to use that power to set up a hereditary monarchy. What he would not be able to do is to retake the Northern Italian city-states as they were geographically separated by the Alps and culturally different from Germany.
The Hohenstaufen dynasty, based in Swabia, would rule Germany for the rest of the 1200s.
Long term effects: With no 4th Crusade (1204), and with far stronger Crusader states, there remains a large Christian territory in the middle east all the way up to the 1350s. A Byzantine reconquest of Anatolia could butterfly the Ottomans out of existence.
The Teutonic knights were founded in 1187 after the loss of Jerusalem and so there is no reason to think they wouldn't exist in this timeline. They would still Crusade in the Holy Land, against the Cumans in Hungary, and against the Balts in the Baltics. The Teutonic state would obviously be annexed by the German emperor for protection from Poland and Sweden in 1386.
The German Emperor would be opportunistic in the Hundred Years war (1337-1453) just like the Burgundians had. He would support whichever side promised him more. But ultimately, he would side with France to prevent the English kings from unifying England and France into 1 united superstate. Burgundy, the Swiss Confederacy, and Austria would only exist as provinces in this empire, not major players in European politics.
With the arrival of Gunpowder, the HRE would become a gunpowder empire, becoming even more centralized.
Entering the 1500s, the Holy Roman Empire has some of the most important lands in Europe. Swiss mercenaries were the most sought after soldiers in the continent playing a key role in the Italian Wars. The Burgundians, in our timeline, had emerged from the Hundred Years war as a major power with a heavily gunpowder army. Bohemia, in our timeline, defeated armies many times its size in the Hussite Wars (1419-1434). All these crack troops would be under 1 banner. The HRE would be so powerful that it would take a Franco-Polish-Swedish-Spanish alliance just to contain it.
Eager to break its continental encirclement, the HRE would ally with Britain and turn to the new world for profit, plunder, and glory. Its too far into the future to know if a Gutenberg Printing press would be invented or if a Protestant Reformation could happen. But my gut says that the Emperor would nationalize the use of the Printing press as a propaganda machine to improve his image and rally the country to war.