Dowels can be quite obvious if you're staining, as you'll get the end grain on the dowel taking up a lot more of the stain than the rest of the body (apart from starting out a different colour). You could slightly countersink the dowel and put a round cap of ash veneer on the top to help it blend in a bit better.

Yes, wood glue should be stronger than the wood - though it does depend on the wood/grain orientation of the join as to relative strengths. Wood glue is certainly stronger than the lignin holding the (much stronger) cellulose tubes together, so when gluing most joints, you do get a joint that is stronger than the wood. End-grain to end-grain still gives a strong joint, but it's not as strong as if the wood was a single piece.

But it's not going to be easy to remove that neck pocket section in a single piece. You can cut down the sides but you'll need to chisel the remaining block off, and the wedge effect of the chisel is likely to end up with the block splitting along the grain. Also, cutting will remove some of the wood, so, certainly at the sides, you may have quite a gap to fill. Better using epoxy than wood glue for that as wood glue becomes weak once you can see daylight between the surfaces to be glued.

I'd still suggest leaving it in place and adding more wood to fill the pocket. I just think it will be less hassle overall. But I've never attempted either method, so I'm only making suggestions based on gut instinct.