If New Testament has κρεμάμενος “hanged” referring to Jesus, why has the word been rendered as σταυρωθείς, “crucified”?

Well, both do indeed occur in the New Testament. “Crucify” σταυρόω is the usual verb, but Galatians 3:13 uses ὅτι γέγραπται Ἐπικατάρατος πᾶς ὁ κρεμάμενος ἐπὶ ξύλου “for it is written: cursed is he who hangs from a pole.”

Galatians 3:13 uses hangs from a pole to refer to Jesus, but in fact it is quoting Deuteronomy 21:23: you must not leave the body hanging on the pole overnight. Be sure to bury it that same day, because anyone who is hung on a pole is under God’s curse. Deuteronomy is referring to death by hanging from a pole. Acts 5:30 also uses that expression to refer to the crucifixion, and the commentaries explain it as an allusion to the same source.

Greek Orthodox hymns generalise this quotation to refer to the crucifixion, on both Holy Thursday and Good Friday: see Επί ξύλου κρεμάμενοι όλοι μας. But translating crucifixion on a cross into hanging on a tree is hardly rare in different cultures. I’m pretty sure it shows up in Old English, though I’m not finding the source on Google.