The Khazars were the alternate silk road which the Byzantiums relied quite a bit on especially when the regime controlling Persia was hostile (which is like what... 90% of the time?) it was for this reason that the Khazar state was even able to form, since they were able to gain massive wealth from the trade passing around the Northern Caspian to the Don River and then into the Black sea and Constantinople. it was also for this reason why the Khazars were generally likely to side with the Byzantiums (who were their customors) than the Persian / Arabs (who were commercial competitors)
From a strategic POV, the Byzantiums were much more at risk from naval invasions from the Black Sea then they were from the Med, since most of their navy and naval bases are in the med anyway. the Arabs learned that the hard way in their initial siege of Constantinople. however in the open black sea the Byzantium navy would have a much harder time blocking naval invasion and raids. hence why they had trouble against th Viking / Rus people later on
Strategically speaking, if the Arabs can hold the Don river they effectivelly shut the Byzantiums out of the silk road completely, plus essentially dooms Crimeia, plus the can raid the northern Anatolian coast and the coast near Constantinople with impunity and also link up with the Bulgarians, it was a win - win situation for them in every way. they simply ran out of man power to pull it off.
Most people seem to grossly underestimate the economic importance of the Ukrain / Russia region to the late dark age / high medieval period. the Golden Horde capital of Sarai was for a period of time probably the largest city in all of Europe, it could only happen because there was a comparablly powerful economy to support it.