Correct. Peter confirmed that David's throne is a heavenly throne occupied by Christ since His resurrection.
Immeasurably more than an illustration. As the One in whom are fulfilled all the promises of God (2 Corinthians 1:20), Christ is the promised Successor to David's throne. As the One who is the Heir of all things (Hebrews 1:1-2), Christ is the Heir to David's throne. Full-on flat-out fulfilled doctrine.
You are missing all the fulfilled dots.
Man made dots do not count. Peter was talking to Jews who just crucified or heard about the Crucifixion of one claiming to be God. Peter pointed out their beloved King David's prophecy, and claimed the Resurrection of Christ fulfilled it. End of story. Ruling and reigning in Heaven may fulfill it in your head. But the audience would never see it that way. Your hindsight does not a doctrine make, 1990 years post message. The Resurrection would prove slightly that Jesus was a risen Messiah. But it would not make David's throne rule the sole purpose. Not even Spiritually if there was no one to Spiritually rule over. No one put any of this together 40 days after the Cross, but perhaps the disciples and a handful of faithful followers.
Peter was back in his boat fishing when the ascension of Jesus and the OT church took place. John 20:8-10
8 Then went in also the other disciple, which came first to the sepulcher, and he saw it, and believed.
9 For as yet they knew not the Scripture, That he must rise again from the dead.
10 And the disciples went away again unto their own home.
John 21:1-4
1 After these things, Jesus showed himself again to his disciples at the sea of Tiberias: and thus showed he
himself:
2 There were together Simon Peter, and Thomas, which is called Didymus, and Nathanael of Cana in Galilee, and the
sons of Zebedee, and two others of his disciples.
3 Simon Peter said unto them, I go a fishing. They said unto him, We also will go with thee. They went their way and entered into a ship straightway, and that night caught they nothing.
4 But when the morning was now come, Jesus stood on the shore: nevertheless the disciples knew not that it was Jesus.
From Scripture, it took several visits from Jesus over the course of weeks for the Resurrection to sink in even for Jesus' closest followers, including the disciples.
Claiming Peter's message, even Spirit led, was about setting doctrine and creating theology, is stretching the limits of interpretation. It was about the Good News of the Atonement and the risen Savior. Peter took a point from David's writings, and declared Jesus as the risen Lord. Any rulership was not literal but figurative. Not that it would never happen physically.
Now we know Jesus was not going to set up an earthly Kingdom. Those in the 1st century were constantly expecting that to happen. It took the writings of Paul to clear that expectancy at point to a heavenly kingdom. But Paradise is not a kingdom, but a city or the address of all Adam's descendants in one place of residence.
Trying to figure out kingdoms, authority, physically in Paradise, or physically on earth is never going to be settled by posters here in 2020. Most live in a democracy of the people and by the people. We can have a head knowledge, but one can only understand until it is experienced. It is hard enough trying to understand that Spiritual is no different than physical. Both are tangible in the same way. Both were created by God as equally part of physical creation. When we recieve our robe of white, we will shine as stars. Adam had a body of light surrounding him before he died, the instant he disobeyed God. Christ will rule on earth for 1000 years as promised to David. That rule will continue on in the NHNE. The proof of that is in Acts 2, and it is called a Resurrection.