I am having the same issues as others in the post and have read the replies with great interest. After I migrated to a new iMac from a 2008 iMac, I wanted to continue my TM backups on the same external drive. After turning TM on and off a couple of times and dismounting the disk I got the "would you like this computer to inherit" option and chose it. After completing the backup ( it seems to have done a complete backup of the new computer) my old backups were grayed out. I fixed that with shift command C and can view all my backup history. The only problem (and it is a big one) is that my user folder denies me access because of permission issues.
Does it show the red "no entry" badges on the folders, something like this?
If so, the problem is, your user account isn't recognized as being the same as the one on the backups. It's not clear from your description what caused it, but see the pink box in Problems after using Migration Assistant for an explanation.
My take on what I have read is that this is quite common and from AppleCare, there is no real solution.
It is common if you don't use Setup Assistant when your new Mac first starts up. Unfortunately, that window doesn't make it clear that it's far better to do it then, not later with Migration Assistant.
There are some workarounds, if you haven't run new backups, in the green box of the link above.
All files that you thinkyou may need should be restored from your backup before migration to the new computer
I'm not sure what you mean by that. If you mean you've been deleting things from your system, expecting Time Machine to keep the backups indefinitely, no, that's very dangerous. See Time Machine - Frequently Asked Question #20.
Changing permissions on the TM is a non starter.
I now question what the utility of inheriting the old backup is if it is not functional for the most important files in my user folder.
If you use Setup Assistant when the new Mac first starts up, or Migration Assistant after one of the workarounds in the green box, that doesn't happen.
If you can't use one of those workarounds, you should be able to access the backups from a different user account. You may have to create a few to get the UID you need.