That is a fundamental misunderstanding of the historical context. Soon after the Arab conquests most of Arabia descended back into raiding tribes and stayed that way until the 20th century. Go watch Lawrence of Arabia.
Most of the other Arabs aren't really Arabs in the sense they are viewed today. The Western Half was Greek and Roman for a thousand years. The eastern half was Persian for nearly just as long. Peoples residing in those areas had thousands of years of 'civilization' as most think of it. It's where it started.
By the time of the Mongols the Arabs guidance of Islam had already receded. Baghdad was the capital of a puppet Caliph. The powers were the Seljuk Turks or the Kurd Ayubids who founded a dynasty in Egypt. Or the Persians. In the Maghreb, it was Berber states. In Spain, it was a mixture. The people's and traditions were only nominally Arab, in the sense they were Islamic and shared that heritage with each other. But their really history is the fusion of the great civilization of the ancient and classical worlds.
The Mongols did not destroy or set back as much as is portrayed. Ok the contrary, the Mongols truly were lenient with cities that submitted and generally adopted the culture and character of the conquered peoples. They were Muslim, Buddhist, Confucian, Tengri, and so on. Despite the claims of reaching from sea to sea because of divine guidance, be real. These people were warring and migrating across the world because life was hard. They were such great warriors because they had to fight for everything they had. Which was pretty close to nothing before they set out in conquest. Brutal, yes, but did they wipe out the culture? Probably not. If wiping out a single city were enough to set the region back centuries, that probably speaks to the nature of their advancement in negative light.
Why had Islam become so advanced relative to the West (they weren't to the East)? It's pretty intuitive. Thousand of years of culture. The knowledge of the Greeks and Romans, the Persians and Mesopotamians condensed into their land. To their credit, they adopted literacy and preserved that knowledge unlike the tribes from which modern Europe largely emerged. Then they acted as a gateway to the East. They deliberately limited the flow of trade and knowledge to control it their benefit. It was profitable of them to do so.
But keep in mind who they were preserving this knowledge from. Themselves. When the Europeans "rediscovered" this knowledge, the Renaissance happened. Columbus discovers the new world, partially to find a path to India to get around the various Islamic states. Portugal sales around Africa and soon arrives in India. The knowledge wasn't even lost to the Islamic states. They just didn't really do much with it because they, as I said, had fractured into regional groups and were every bit as concerned with fighting each other as the Europeans were. Much like the Byzantines shielded Europe from Islam, the Turks shielded the Islamic world from Europe. As soon as it finally fell, the Islamic world was gobbled up from Malaysia to Maurerania to West Africa.