Depends.. when your done,what would change? I grew up in the oak cabinet craze and can hardly stand it. That being said, I have honey oak cabinets in my house. Refinishing isn't an option for me. Dyes and stain may hide it but it's still red oak under there and I know it.
Painting would be fine. I'll build some some day.
One tip would be to start with longer than you need. It is not uncommon for the tap to take an inch or two to square up and be cutting right. Wax is good for a lubricant. Just don't advance the cutter too much. Many light passes gives a far better result than trying to hog through it in two or three goes
Most people in the forum use west system epoxy. I haven't used it in this manner, but I have seen a number of youtube videos where people sand the epoxy surface along with the wood all at once, and it cleans up very well.
I will add a link if I can find one in the next 30 seconds.
Here you go, here's one from Matt Cremona..I would probably just follow his steps for adding dye (I have transtint dye as well, that's a good option, and they have lots of different "colors"):
West System for the epoxy. It's not really going to matter which hardener. Even the 205 "fast" hardener is slow compared to say a 5 min epoxy.
I've used mixol universal tints and transtint dyes with success. A little goes a long way with either.