How do I get permission on my own Mac Studio to store and transfer files as I want them?

Put them in your home folder or /Users/Shared/ instead of the top level of the drive.

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To help you understand what's going on, since the advent of Big Sur, one of the changes to macOS security is that you can no longer create a folder or file at the top level (aka root) of Macintosh HD. Having an admin account makes no difference, the root level is also secured against admin accounts.

All user created folders & files must be put somewhere inside your user folder (aka home directory ... as in /Users/<yourusername>). Alternatively, you could, as @etresoft posted, create an additional Volume, which you could treat as if it were a separate disk drive including creating folders/files at its top level (root level).

If you want an experience that is closer to what you were used to before, you can try this:

1) Launch Disk Utility.

2) Go to the toolbar and click the "+" button above "Volumes"

3) This will create a new volume on your hard drive. You can completely organize it the way you want - no restrictions.

Caveats:

Just make sure that this volume is getting backed up. Since it isn't in your home directory, it might be excluded by default in some cases.

If you want to use iCloud, you will still have to use folders in your home directory.

Your Desktop will still be on your home directory.

Various apps will still store there data in various folders in your home directory.

Remember that you've done this. You don't want to accidentally delete this new volume one day without having a backup.

Have you tried setting Ignore Ownership of the EHD?

The way you've been doing it won't work for the volume containing macOS anymore. From the internal HD, open Users and then either the folder with the house icon or Shared.

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