I glued with Titebond, let it sit overnight, and slowly hand sanded with 60 grit, and then finished with 100 grit (for now).
A number of people seem to use a drill press as a router. I would be worried that they don't get enough RPMs.
I have a Dremel and another off-brand rotary tool. I confess I don't control them as well as I would like. And if I went at it with a coping saw I might be too old to care when I finish.
The nice thing about a router is that if you use a template they are fast, precise and require very little skill. Caution, yes, but not skill.
I don't use MDF anymore. Too succeptible to FL humidity. I mostly use furniture grade plywood. What I used this time was some left over door casing, which worked well. Easy to cut and shape to a smooth edge.
The problem is the grain and the headstock. There was a little burr where two cuts with a jigsaw came together, right where material was very narrow. The the blade knocked the little tab off instant it hit the burr.