I'm no expert (and given that this is my first post, many might be tempted to completely ignore my opinions) but... at 50+ years old and having had a grandfather and father that made their own tools so that they could then use those tools to make bigger tools and workbenches, I was raised on using stones to sharpen everything. And each tool often had its own specific stone (and God help me if as a kid I used the wrong stone on the wrong tool!!). I was whupped more than once for causing irregularities in one of my Dad's stones... I still have many of them today, and consider myself pretty good with nothing more than my eyeballs, hands, and a stone on a flat surface. That being said, I sharpen a LOT of different kinds of edges - everything from spearfishing "iki jime" and shaft tips to single-edged sushi knives, folders, hunting knives, boat fillet knives, and antique Wustof/Henckles kitchen knives. They all have wildly different profiles and no one tool is "best" for them all. And I find that many new knives I buy need to be re-profiled anyway...
In addition to my carbide, diamond, and Japanese stones, I also have an Apex Edge Pro. At first I really liked the Apex but I got really tired of having to constantly flatten and replace the little stones. I also found that if I got lazy or sloppy, (or didn't have a perfectly level work surface at the right height) the results were about as inconsistent as doing it by hand. There is a running joke in my family about Dad spending entire Saturdays at the dining room table sharpening knives, and part of the reason is the lengthy process of re-profiling knives with the Apex. For this reason, I broke down and recently purchased the Ken Onion edition of the Work sharp. I love the fact that the belts never develop a "curve" that will throw off the edge profile - they just become less effective at cutting that edge. I also love how FAST I can re profile an edge when necessary.
I agree with all the comments about it being *really* easy to destroy a knife with a belt grinder/sander. But that risk is precisely the main advantage when in the hands of someone who is careful and deliberate about setup.
At this point, I rarely use the Apex any more, and only use my stones for touch-up or field maintenance when on a hunting/camping/fishing trip.
My vote is for a belt sander but with the caveat that it is better used with careful understanding of what grit belts to use and using a guide for reliable/repeatable angles.