Why macOS asks for permission
The folder chooser is not a bug, it is the only way in.
The short version
DiskPeek is sandboxed because the Mac App Store requires it. There is no entitlement that grants an app free run of your disk, so the only way it can read a location is if you pick that location yourself. That is what the folder chooser is doing.
Full Disk Access will not help
It is the obvious thing to reach for and it genuinely does nothing here. The sandbox refuses before macOS ever consults that setting, so switching it on changes nothing. Grant the folder itself instead.
What stays out of reach
Even inside a volume you have granted, macOS keeps some things private: other apps’ data, Mail, Messages, the Photos library, and other user accounts. Those cannot be granted to any app at all and will always show as restricted.
On a full drive scan a few hundred restricted items is normal. On a recent scan of mine it was 639 out of nearly six million, about one hundredth of one percent.