Tashman,
Don't use Super Glue. It does work on porous materials, and can be made to work on wood, especially if you have an accelerator spray, but I've never had good luck with super glue for gluing wood to wood. I do, however, use CA glue (Krazy glue, actually) to glue Ivory keys covers back onto the wood piano keys when I'm out "in the field". It works like a charm IF both surfaces are clean and level.
And that's brings me to my point. All of these comparisons of Gorrila Glue and Titebond and Elmers and WHATEVER. All of them are done under ideal conditions: new clean wood joined to new clean wood, clean smooth surfaces, and all joints tightly and properly clamped for the proper amount of time. And even the weakest of them is incredibly strong under these conditions.
THAT is far more important and vital than exactly what TYPE of glue you use. Elmers white glue is more than adequate to your needs here IF you have a joint that isn't contaminated by old glue or dust and grit, and which mates back together well, AND if you CLAMP THE JOINT WELL AND GOOD. That's key. Short of using epoxy, you must ensure good and proper pressure on the glue joint while it is drying.
In your case, tightly wrapping the handle with first-aid tape or string, using multiple wraps, tied off tightly and securely, would ensure a nice even pressure I think. Piano technicians sometimes use this technique to reglue broken hammer shanks on a piano. I kept a thing of dental floss in my kit expressly for this reason.
If you can't keep good pressure on the joint for some reason, then epoxy is the glue of choice.
My personal favorite glue for wood, ivory, felt, leather, and other organic and porous materials is hide glue. You have to have a glue pot to maintain a water bath at the right temperature, and you have to buy dry crystals and mix them with the right amount of water. And you have to get a feel for the right consistency of the glue. And it smells just a bit like wet dirty socks. But it is the most widely used glue in a piano action, and I've spent hundreds of hours next to a hot glue pot, gluing felt or leather to piano parts. It's an amazing glue because it tacks very quickly, but then takes about 12 to 24 hours to fully dry. It holds stuff like felt and leather to wood very well with only a handful of seconds worth of initial clamping pressure, but if you glue wood to wood, and you clamp it well and for 24 hours, it is pretty much as strong as the strongest of the modern glues. Plus, it is completely reversible. Get the joint wet, add heat, and it will let go. This is very important for things like piano key bushings where you want to be able to re-do the bushings every decade or so. Use white glue or gorilla glue or pretty much ANY modern glue, and you will only do it once, and the next time you will have to painfully and somewhat destructively, remove the glue residue, assuming you can remove the felt or leather in the first place!
This is why people like piano tuners and violin makers and guitar makers and furniture repair people HATE Elmers white glue. It's ubiquitous. It's everywhere. And people use it to glue up joints without cleaning the joint and without clamping the joint, and so it doesn't hold well anyway and makes a complete and utter MESS of the joint for anyone who comes after. If a chair rung has been "reglued" with elmers in this way, your only hope is to pull the thing totally apart and painstakingly scrape off all the white crap until you get down to bare wood. Then, and only then, do you re-glue, with inventive and ingenius use of ropes and twisting sticks to tension up the joint.
Anyway, I'm ranting. The point is that the PREP-WORK and TECHNIQUE you use to glue are generally a lot more important than the TYPE of glue you use.
