Just a few comments on some of scholar's claims:
The simple facts are that Jesus died on a stake because this is what the eyewitnesses saw, they saw a stauros which means a pole , stake or whatever. The testimong does not suggest a stake with a crossbeam or another piece of wood affixed to it, they did not see two or three or four pieces of wood but a simple, single piece of wood described by the word stauros. It is very, very simple.
Obviously scholar has not read Leolaia's demonstration that stauros can mean both the patibulum (the stauros carried by Simon of Cyrene according to the synoptics and by Jesus according to John) and the complete instrument (the stauros to which Jesus' hands and feet were nailed).
Also, Jesus could only have died in this mannner because under Jewish Law, criminals were hung on a stake or stauros so Jesus died according to Jewish custom but under Roman authority. Jesus's death had to fulfill Jewish Law completely.
Killing somebody by hanging him on a cross -- or stake, for that matter -- was no established Jewish custom, and certainly no Jewish law. In spite of later Qumran interpretation, Deuteronomy 21:22f originally refers to hanging the corpse of a dead convict (previously killed by stoning which was the common penalty for blasphemy -- the charge which the high priests leveled at Jesus according to the Gospels).
Moreover, John 18.29ff clearly implies that the kind of death Jesus was to suffer was the Roman one:
So Pilate went out to them and said, "What accusation do you bring against this man?" They answered, "If this man were not a criminal, we would not have handed him over to you." Pilate said to them, "Take him yourselves and judge him according to your law." The Jews replied, "We are not permitted to put anyone to death." (This was to fulfill what Jesus had said when he indicated the kind of death he was to die.)
Besides these simple facts without engaging in the historical debate there is the problem that the cross is a pagan symbol and it would be a violation of the sacred for Jesus to be associated with a pagan thing. Jesus death was sacrificial and a sacred act of atonement and it could not violated by the introduction of something unclean and detestable.
It has already been shown that a stake or pole is a much more common "pagan symbol" than the cross. Moreover, the point in Galatians 3:13 is precisely that Jesus' cross is a sign of divine curse:
Christ redeemed us from the curse of the law by becoming a curse for us--for it is written, "Cursed is everyone who hangs on a tree"
This text quotes Deuteronomy 21:22f which uses the technical vocabulary of ceremonial uncleanness:
When someone is convicted of a crime punishable by death and is executed, and you hang him on a tree, his corpse must not remain all night upon the tree; you shall bury him that same day, for anyone hung on a tree is under God's curse. You must not defile (Heb tm', to make ceremonially unclean) the land that the LORD your God is giving you for possession.