The issue in cutting rabbets (ledges) for the binding occurs on instruments whose tops and/or backs are not flat. If the base of the router sits on the curved/arched top or back, the binding ledges that are created are not of uniform thickness. The angle of the router bit relative to the instrument sides changes as the router base tilts while traversing the circumference of the arched top or back. This creates ledges of varying thickness - thinner the further you are from the center of the arch.
If you are only making one or two instruments, it is easy enough to just use the router with its cutter to create the rabbets. Follow that by using a sharp chisel or files to make the rabbets uniform in thickness. If you do not, when the binding is glued and then scraped flush with the sides, the thickness of the binding will vary.
If you are to make a number of instruments, you may wish to invest in jigs that will ensure that the sides of the instrument are used as the reference for the router. As long as the base of the router is maintained perpendicular to the instrument sides - that is, the cutter is parallel to the sides - the thickness of the ledge will be uniform. As you have seen, there are numerous ways to achieve that, from attachments to Dremels to large holding fixtures. All of them will work.
Regardless, pay particular attention to the sequence in which you route. In the "parlour guitar build" video you posted, he shows this about half-way through, though doesn't explain why. He routes from the widest point in the upper bout towards the neck, from the widest point in the upper bout towards the waist, then from the widest point of the lower bout towards the waist, then from the widest point of the lower bout towards the end block. This is done so as not to "catch" the wood and sever it along a grain line. If the wood does "catch", it can take a 1/4" or 1/2" piece of the top or back when it does so. You can reglue the piece, but it is best to avoid doing so to start with. With the amount of overhang you have, nibble away at it, rather than try to remove it in a single pass of the router. If you have access to a disk or belt sander, I'd remove nearly all of the overhang that way. A sharp chisel will also work, if you have that skill.
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