We have tons of time shares here, being in Virginia Beach and Williamsburg.  The company is correct, they are filed just like a deed.  The legal description reads as most do with additional information of the week number of the unit being conveyed.  People even mortgage the time shares.
But if this company wants to transfer title of the time share to their name, they need the unit owner to sign the deed transfer, not the time share company. 

After the deed is recorded, the Clerk sends back the originals to the title company who recorded it, or to the owner if the time share company recorded it.  Our Clerk's do not hold on to originals. 

Maybe I am too use to this and I do not see the new idea here. 
Is this company working for the time share company.