That's a shame.
Never mind, take a positive approach and think that perhaps there was packet loss during the GT6 update download process and you needed to do that anyway.
For future reference, here's the story.
Clearing the cache in-game only deletes temporary data downloaded form the network.
http://www.gran-turismo.com/au/gt6/manual/#!/option/network
Deleting the installed files
in-game in GT6 simply means slowing down the load times for a while.
It will be forced to read data from the disc as opposed to reading the same files copied to the HDD.
But there is a chance the copied files are bugged, so it's a viable solution.
Safemode isn't something to be worried about.
It may take a little time to go through the process, but there is nothing to fear.
For the 2 options I've suggested, almost nothing will be removed or reset, but the video explains the situation.
For a disc based version, all update files are kept in the PS3's Game Data Utility folder.
Deleting this file means you are back to version 1.00.
For a digital download, the update files are added to the files that have the actual game in the GT6 game icon.
You'll need to delete the entire game and start again to delete any update files.
You will need internet access to sign in to the PSN before being able to update the game again.
If your PS3 is able to run the game, but it freezes at trying to enter your name you have another issue.
It sounds like you use a disc version.
It seems you've passed this point in the game before, so something has changed.
Is the disc clean?
Also, seeing as you have no access to the game as it stands, I would (if you like, copy your save to USB first) remove all evidence of GT6 from the console and start again.
This means deleting the GT6 file in the Game Data Utility folder, and both the main and back-up save files in the Save Data Utility folder (your issue could be a corrupt save file).
The game should run successfully as it comes on the disc (V1.00).
Edit
'All of your save game data'

I hope that wasn't advise from a support rep.
Ill-advised is the kindest thing I could say about that advise.