Can an LLC formed solely to own and operate an aircraft be operated under Part 91?
The short answer is yes, an aircraft owner can hire a pilot to fly him or her, as well as other friends and family, and operate under Part 91.  This is called private carriage, as opposed to common carriage.
Just to clarify a few points:

The pilot hired would need to have a Commercial Pilot Certificate, even if no money changes hands, because the FAA considers flight time as compensation.
The owner could not charge other passengers because this would constitute common carriage, (operating as a charter) and require a Part 135 Operating Certificate.
It makes no difference whether the owner is a rated pilot or not.  Some owners fly themselves, but want the security of having a more experienced pilot along with them.

P.S.  I realized looking related questions on the right side that this is pretty much a duplicate of this question:  Can an aircraft operator hire a Commercial Pilot to fly their aircraft under part 91?