What else would you consider an absolute basic necessity in woodworking tools, for building a neck-through LP basic style guitar?
I imagine I'd get a pre-slotted neck, so that's one challenge I'd dodge the bullet on (*whew*)
I imagine I wouldn't be building many, to start... I'd start with just the one, and decide whether I enjoy the building enough to make more, or if I'd just enjoy the one and only.
I also plan on possibly building a tube amp, like they do on this site, among others. I'm thinking a 15W tube amp, with some killer overdrive tones.
I built my first electric solid body in my lap for the most part. I hogged out the cavities with a drill motor and then cleaned it up with a chisel (it is also chambered) and has a 1/2" maple cap.
I glued up the body by putting it on a scrap piece of plywood and nailed on some 2x4 scrap, then used some wedges to clamp the pieces guitar.
I cut out the body outline with a saber saw. I used a pocket knife to make the carved top and I sanded alot.
It turned out wonderfully with only one noticeable flaw and then only noticeable if I point it out.
Most of the shaping was done with a pocket knife.
I did it with these minimalist tools because that is what I had. However, I would not hesitate to order from Warmoth one of their body blanks that have the hardware and pickup cavities routed. Then shape with a saber saw, use a pocket knife or any kind of knife I could find to shape the top and bevels and hand sand. I say build guitars with what you have and the rest of the equipment will come, but Randy Parsons who builds for people like Jack White some of the most advanced electric guitars made uses very minimalist equipment, mostly hand tools
so to answer your question, none of the tools listed are needed to build electric guitars.
I suggest that if you have friends with tools you might be able to get their help.
I have a jointer from the 30's that I haven't used yet (just got it)
Paid $25 for it without a motor and I have a motor for it.
The point being if you can do without it, do without it till an orphaned tool makes a home with you.
If the lack of a tool keeps you from building, then the problem is not the tool
I built my first electric with nothing more than a cheap scroll saw, a hand drill, a screwdriver, and some sandpaper. Not that I would do it again that way now that I have better tools, but you really do not need a whole lot of tools to get started.
Based on my experience so far, all of the tools you listed will come in handy but what you REALLY need is a good hand plane, a good straight-edged metal ruler, some good quality and very sharp chisels, the band saw, and a good eye. If you're good with a jig saw you wouldn't even need the band saw but it sure is quicker and easier. That's it. A joiner comes in handy, as does a planer, but you can use a drill press and chisel for most of the things people use routers for and quite few people (including me) cut the truss rod groove with a table saw not a router.
You will be amazed at how much you can do with one good hand plane and one good really sharp chisel. I know I was.
I'll second the "scary sharp" technique. The planes and chisels I've been using were sharpened that way and they carve through mahogany easier than a sharp knife through butter and carve maple almost that easily. The hard part was learning not to push too hard!
Did I mention a scraper? I don't know how much they cost (I'm using borrowed tools) but I am in love with the one I've been using.
Scrapers are only a few bucks a piece, cheaper if you make your own. Dirt cheap compared to sandpaper, they do a better job, and produce no sawdust to sneeze at. One of the most underrated tools, IMHO.
When I started making electric guitars, I used a jig saw, drill press, and files and sandpaper to make jigs and templates for all my parts, and cut everything with a router. My first was cut with a jig saw, but I had a bit of sanding to do.
To do a neck-through LP, though, you need a way to accurately dimension wood, in thickness, length and width. Whether it be bandsaw and hand plane, tablesaw and planer... You'll need a way to carve the top; whether it be chisels and violin planes; archtop jig, etc. You have to make electronics pockets; drill and chisels, router and templates. Rabbeting bit or such for binding. You need holes for hardware; drill and assorted drill bits. Fret slotting saw. A way to do the inlays (scroll saw, jeweler's saw, Dremel, small chisels). Rasps, files, spoleshaves, sandpaper. Assorted clamps. Soldering iron for electronics. Screwdrivers (and nut driver for Gibson style trussrod nut). Sockets or wrench for tightening jack and pots. Spray equipment and related items for finishing. Good light source. Extension cord. Garbage can for debris (and mistakes).