GTA 5 – How to Fix the Replay Mission Big Score Freeze Bug at Door

If you are unable to destroy the car before the mission, and can't move it, you could try killing yourself when you destroy the garbage truck - if you fail a story mission often enough, you get the option to skip it.

So I kept putting off the whole uninstalling the game and resetting the console because it was a pain and played something else. However, the problem was fixed when I went back to play the game at a later date and there was an update. I'm guessing this refreshed the install somehow in the same way that reinstalling the game would and that there had also been a console update which had reset any cached information.

If anyone does try the "official" method listed above and finds it to work (rather than waiting for updates), then write it here as an answer and I'll change the accepted answer to be that one.