Is it possible to have full hard drive access with a snap?

Yes, what you want is an interface that grants full access to a disk, and it's reasonable for such an interface to exist. It would not be an auto-connecting interface, in that either the user or the person who makes a device with the snap would need to explicitly tell the snap system to allow that connection to take place, but the interface itself is a reasonable request.

If you're game I'd suggest you work up a patch for snapd which we will shepherd into the code.

Since I posted my question, Snapcraft has gained a new confinement option: "classic".

From: https://docs.snapcraft.io/snap-confinement

Classic Allows access to your system’s resources in much the same way traditional packages do. To safeguard against abuse, publishing a classic snap requires manual approval, and installation requires the --classic command line argument.

Duplicity now has classic confinement approved and we have started building snaps in that way.

(See also: https://github.com/snapcore/snapd/pull/6436 which is about potentially adding a new interface for backup applications that only require read access to the whole filesystem.)