This is a question that seems to frequent the various online guitar forums. First of all, to be semantically correct, the entirely handmade description does prohibit the use of power tools. However, I think most people use the term to imply that an object is carefully crafted by an artisan rather than mass produced in a factory. The majority of the work I do is powered by my own strength rather than electricity. I use an array of chisels, files, rasps, saws, scrapers, and planes to prep the various parts for assembly. However, I also employ an assortment of power tools for specific operations and also to do the basic prep work for most aspects of construction. Some of my processes incorporate mostly hand tools and others mostly power tools. For example, in the manufacturing of the headblock and tailblock, I use a bandsaw, disc sander, and drill press for most of the work and only use a small block plane to fine tune the surfaces. For the neck profile, I use a bandsaw and a shaper to dial in the neck thickness and then use a spokeshave, chisels, rasps, sandpaper, and scraper to carve the neck, heel, and volute.
Regarding CNC equipment, I currently use this equipment only for the creation of my fretboards and this process is outsourced. CNC equipment is a bit of a mixed bag. On the one hand they are incredibly useful but they also substitute computer/programming skill for woodworking skill. There are certain things that CNC's can do that no human can accomplish. For example, a CNC can slot the fret locations to incredible accuracy and also have the fret slots follow the radius of the fretboard thus producing a stronger fretboard. They are also incredibly useful for inlay work (rosettes, logos) and template/jig creation.
Both power tools and CNC's create new possibilities but also lower the woodworking skill needed to manufacture a component. Most power tools still require a tremendous amount of technique to use and I personally view them as a great way to speed up the process while still retaining a high degree of skill to use effectively. CNC's are wonderful tools but you absolutely lose a human component with their incorporation into a process. I believe they are invaluable for certain part manufacturing and inlay (which is quite expensive when done entirely by hand) but also believe they can turn parts of the build process into assembly work. Nevertheless, I think it is important to note that even when CNC's are incorporated extensively into a luthier's build process, there is still a large amount of hand work that must be done. The fretwork, shaping the tone, the binding process, and all the gluing processes require a high degree of woodworking proficiency. In conclusion, I think that all of these tools have a place in the shop of builders but I hope that the use of automated tooling is selectively utilized and never entirely replaces the use of hand tools.