Yes He could've. In fact he could have saved us all without the need for Christ, the 2nd person of the Trinity coming to Earth as man. But He didn't. He chose to assume humanity to unite us with His Divinity
That act, echoes through eternity. It's why Lucifer and his followers proclaimed non serviam . Raise man above us pure spiriitual beings? Never.
Humanity has a role to play in salvation history. I know this isn't shared with the TULIP crew but its true nonetheless. The Catholic Church is God's way of doing that. So when we "take credit" as you put it, all we're doing is glorifying God's Divine plan which includes our separated brothers and sisters.
No, He couldn't have. If there were some other way to save mankind, God would have done it.
Sin cannot be in the presence of a holy God. Death is the penalty for sin. Without the shedding of blood, there is no forgiveness of sin.