Why did Kirk's Bird-of-Prey fly slow at Warp 9 for time travel in Star Trek IV?

This was meant to be a reply to someone on YouTube. The question is interesting, because it posits, that at high warp, the ship would have overshot the Sun very quickly.

The warp bubble is typically static, and the ships inside of it do not move, while the bubble is at warp. So, I surmised, that the movement could have been slower, because of the following hypotheses thayt I pullled right out of my head:

(1) They had to move the ship inside the warp bubble, and/or

(2) that high warp was used not just to maintain the warp bubble and still move it at quite high speed relative to the rest of the space, but also to keep it stronger to protect the ship from the Sun's heat and massive gravity, and to extend the bubble's length or depth across the trajectory and through the Sun's Corona in order to make the jump.

Therefore, the expended energy was Warp 9+, while the actual velocity was slower, as the distance inside / depth of the bubble was greater than the pre-Warp-speed distance between the ship and the Sun. This was probably the crux of Spock's computations.