Why did a priest have to arise after the order of Melchizedek rather than after the order of Aaron?

A priest had to arise after Melchizedek because Gentiles could not approach God through a Jewish high priest; they need one representing Gentiles, as Melchizedek did. Also, the Lord Jesus sprang from Judah, not Levi, so He was not eligible for the Aaronic priesthood.

Les explains that if perfection came through the Levitical priesthood, there would be no further need for another priest to rise after the order of Melchizedek rather than being called after Aaron. The answer is that Gentiles could not approach God through a Jewish high priest; they must have a high priest who represents them, and Melchizedek was that representative clear back in 2000 B.C. Les also points out that the Lord Jesus sprang from the tribe of Judah, not Levi, so He was not eligible to be a priest after the order of Aaron. Instead, He is made a priest after the similitude of Melchizedek, not after the law of a carnal commandment, but after the power of an endless life, a priest forever. This connects the high priest of the Gentile with the Most High God from start to finish.

Hebrews 7:11

If therefore perfection were by the Levitical priesthood, (for under it the people received the law,) what further need was there that another priest should rise after the order of Melchisedec, and not be called after the order of Aaron?

Which tribe did all Israelite priests have to come from?

All Israelite priests had to come from the tribe of Levi. Beginning with Aaron as the first high priest, no one could serve as a priest unless they were of Levi. The Levitical priesthood was therefore the required lineage for Israel's priests.