2. If someone today believes they are the son of God or Satan or whoever, no one would believe them. They would be treated for mental illness.
The title "Son of God" has nothing to do with illusion of mental illness. It has historical basis in jewish world. It was a common term in the jewish theology and enthronement rite and not a term for a biological son of god but a honory title.
The titel "Son of god" appears in the old testament of the hebrew bible already before Jesus Christ lived at all.
Hosea 11,1 calls the people of Israel "the son". In the Psalms the enthroned kings are called "sons" of god. Of course the jewish kings did not think that they were biological sons of god, but adopted the ancient oriental tradition where the at the kings courts the honory titel "son of God" was used at the enthronement ceremonies. So the term came to Israel from oldoriental, babylonian enthronement rites.
Psalm 2 uses this oriental royal court language in the enthronement scene. But here in Psalm 2 with this formula the king is not the biological son of God but is chosen and adopted and gets obligated legally to act as minister in the place of god, representing god. The kings were also anointed with anointed oil, like christians today do at the confirmation, what underlined this ministry.
6 Saying: “I myself have installed my king+On Zion,+ my holy mountain.” 7 Let me proclaim the decree of Jehovah;He said to me: “You are my son;+Today I have become your father.+Psalm
The king David was Son not because he was it biological, but because he was adopted as son, in the language. Therefore the PROCREATION, another term, is the act of chosing by god. Israel is likewise called the chosen Son of God. The kings are chosen, the anointed are chosen ones. - There is no biological procreation.
With that in mind the term "Son of God" was alread well known in the jewish world and should not mean that Jesus is the biological son of the father but rather it meant a king or Messiah.
But of Jesus is told that he was killed because he agreed to be the "son of God". It was the blasphemy for what he was killed to say to be the son of god. Was it because no one else was allowed to use the title than the real king the awaited Messiah.?
Of course it could be that a wannabe-Messiah who wanted to become king or messiah lived and was killed and that he was so ill that he said he was even a biological son of god and not that Jesus he was like in the gospels a great teacher a intelligent rabbi.
or it could have been that the story of a Wannabe-messiah was only a creative invention of the later church to have a hero, but he was in reality a looser, but who wants a looser, a crucified, as founder of a religion, the romans who wanted to defeat the Jews?
the dead sea scrolls dating about 200 - 100 BCE often combined the title "Messiah" with Son of God and Son of the highest. Another fact that shows that it was a common term.