I hear what you're saying, but take the Mongols. They had what you call "less advanced" cavalry, but defeated everyone. Or consider the USSR. They had essentially primitive weaponry, but lots of it, so they won. Isn't whatever leads to victory "advanced". It's a hard question!

Just some points. The "massive numbers" of expeditionary forces sent by the Hungarians doesn't seem to have ever exceeded 5,000 and they were very strong in defence. (907, 1030, 1051/2)

The Magyar bow was more advanced than anything in Europe and would remain so for centuries.

They also kept mounted archers until advances in gunpowder pushed archery out everywhere around 1500.

You point about the Europeans squabbling is true, and the Magyar expeditions were very carefully calculated to keep it that way. Are you aware that in every expedition, they were in alliance with the "other" king, or prince or whatever? Hungarian historians have now, after some 30 years of debates, rejected the idea that these were raids. The current view is they were centrally controlled military campaigns aimed at keeping Europe divided and avoiding the fate of the Huns and Avars. Seems like the plan worked!