This is my best guess given what are written in the Bible and what are consistent with Jesus, Mary, and Joseph's characters.
Luke 2:41 Every year Jesus’ parents went to Jerusalem for the Festival of the Passover.
This was an annual event with set habits and customs.
43 After the festival was over, while his parents were returning home, the boy Jesus stayed behind in Jerusalem, but they were unaware of it. 44 Thinking he was in their company, they traveled on for a day. Then they began looking for him among their relatives and friends.
In previous years, when the festival was over, Jesus' parents had no need to make sure that He had departed Jerusalem with them at the same time. They just assumed that He was either with their relatives or friends traveling in the same direction. They had complete confidence in the boy Jesus.
However, this year was different. Jesus didn't depart with any of them.
The question is: Why not?
Meanwhile, Jesus was engaging the Jewish teachers, totally absorbed in the discussions of the Scripture. They were surprised at His young intellect. Jesus felt at home in the temple.
46 After three days they found him in the temple courts, sitting among the teachers, listening to them and asking them questions. 47Everyone who heard him was amazed at his understanding and his answers.
Now, back to trustful parents, after a day, Jesus still hadn't checked in with them. They started to worry and looked for Him. He was not with group! Now they started to panic and headed back to Jerusalem.
Well, you know the rest of this story. Now I am going to answer OP's good questions directly.
Did Jesus go to the temple BECAUSE His parents forgot Him and therefore wanted to remain busy?
No, His parents didn't forgot Him. They took for granted that He was with relatives and friends.
No, Jesus stayed at the temple because he felt at home there. He felt purposeful there with the Jewish teachers.
49 “Why were you searching for me?” he asked. “Didn’t you know I had to be in my Father’s house?”
Did Jesus go to the temple BECAUSE that was His plan from the beginning?
No, it was spontaneous. He would not premeditate an act that could worry His parents.
What triggered the separation between the parents and the child?
They trusted him. They had complete confidence in Him. He had always been obedient. The temporary separation was no big deal in previous years.
51 Then he went down to Nazareth with them and was obedient to them.
Why did boy Jesus, 12 years old, remain behind in Jerusalem?
To mind His Father's business. It was the will of the Father.
John 2:17 His disciples remembered that it is written: "Zeal for your house will consume me."