I have done innertubes the same old way as your dad did. Pliobond (which contains neoprene monomer -- at least some versions did) will also work. I used CA plus a good knot on the bungee cords for my model. It has held up well over the years and been exposed to quite a bit of oily exhaust. Also available at hobby shops is a modified CA for "rubber" -- I suspect is just a slow CA modified with neoprene monomer. I got pretty much the same effect premixing some Pliobond with slow to medium CA, but that is not what I used for the bungee cords.
It is really important that the rubber be clean. I use MEK for that, a little abrasion and then MEK again. Adhesive for innertubes has a sulfur odor, and if you have it, I would consider it too. I believe the sulfur is to improve its adhesion. Many rubbers rely on sulfur compounds for polymerization. As for overlap, given a width of X, I wouldn't consider any overlap less than X. An overlap of 2X to 3X is probably better. Of course, if you rely simply on overlap, the ends will peel away from the base. That is why I used a knot and a tiny bit of CA
Any rubber that is stretched will degrade faster than one that is not stretched. So, if you decrease the amount that you stretch it, that will help. Bottom line, if an innertube will give enough stretch, that is what I would use. I only went to the alternatives because I wanted less stretch and because I did not have an open end to get a closed loop over.
John
Edit: For power drive belts, the overlapped ends are scarfed (tapered). That reduces the stress concentration at the ends of the overlap, but is probably a detail you can ignore.