John 19:1-23 is an overtly political passage about Jesus's crucifixion, which suggests that Rome (or at least Pilate) saw Jesus in political, not spiritual terms. In this passage, Pilate shows a complete lack of concern about Jesus's spiritual claims, but focuses on the political connotations of Jesus's claim that he is the Messiah.
1 Then Pilate took Jesus and had him flogged.2 The soldiers twisted together a crown of thorns and put it on his head. They clothed him in a purple robe 3 and went up to him again and again, saying, “Hail, king of the Jews!” And they slapped him in the face.
This is political humiliation that Pilate and the Romans inflict on Jesus, not spiritual or religious. They clothe him with political symbols. They mock him as the "king of the Jews," which is overtly political, not spiritual language.
4 Once more Pilate came out and said to the Jews gathered there, “Look, I am bringing him out to you to let you know that I find no basis for a charge against him.” 5 When Jesus came out wearing the crown of thorns and the purple robe, Pilate said to them, “Here is the man!”
6 As soon as the chief priests and their officials saw him, they shouted, “Crucify! Crucify!”
But Pilate answered, “You take him and crucify him. As for me, I find no basis for a charge against him.”
The question here is, who has power? Who has authority? Pilate is willing to let Jesus go, but receives pressure from the Jewish religious authorities to go crucify him anyway.
7 The Jewish leaders insisted, “We have a law, and according to that law he must die, because he claimed to be the Son of God.”
8 When Pilate heard this, he was even more afraid, 9 and he went back inside the palace.“Where do you come from?” he asked Jesus, but Jesus gave him no answer.
Why is Pilate afraid when the Jewish leaders tell them that Jesus claims to be the Son of God? Is it spiritual? Is it religious? Is it political?
10 “Do you refuse to speak to me?” Pilate said. “Don’t you realize I have power either to free you or to crucify you?”
11 Jesus answered, “You would have no power over me if it were not given to you from above. Therefore the one who handed me over to you is guilty of a greater sin.”
Again, the question -- who has power? Who has authority?
12 From then on, Pilate tried to set Jesus free, but the Jewish leaders kept shouting, “If you let this man go, you are no friend of Caesar. Anyone who claims to be a king opposes Caesar.”
Again, this is overtly political. Jesus's claim to be the Messiah is not just religious, but also a political one. In order to pressure Pilate, the priests overtly and directly connect the crucifixion to Rome's imperial concerns. They argue that Jesus's messianic claims are not just spiritual, they are disloyal to Caesar. They pressure Pilate with the claim that if he doesn't crucify Jesus, he's committing treason against Caesar.
13 When Pilate heard this, he brought Jesus out and sat down on the judge’s seat at a place known as the Stone Pavement (which in Aramaicis Gabbatha). 14 It was the day of Preparation of the Passover; it was about noon.
“Here is your king,” Pilate said to the Jews.
15 But they shouted, “Take him away! Take him away! Crucify him!”
“Shall I crucify your king?” Pilate asked.
“We have no king but Caesar,” the chief priests answered.
Again, that's political, not theological language.
19 Pilate had a notice prepared and fastened to the cross. It read: Jesus of Nazareth, the King of the Jews. 20 Many of the Jews read this sign, for the place where Jesus was crucified was near the city, and the sign was written in Aramaic, Latin and Greek. 21 The chief priests of the Jews protested to Pilate, “Do not write ‘The King of the Jews,’ but that this man claimed to be king of the Jews.”
The sign above Jesus is a political statement. The priests want Pilate to make it less political, but he refuses.
22 Pilate answered, “What I have written, I have written.”
That’s Rome making a political judgment. While Pilate may not have believed Jesus was a serious threat, his language (capitalizing "King") shows that he's thinking about Jesus in political terms, not spiritual or theological ones.
So I don't think it's correct to say Rome wasn’t concerned about Jesus’s claims until long after his death. The political dimensions are right there in the text.