Brian it kind of sounds like you are not getting the burr removed to me. A edge in my opinion should last longer than 10 cuts on a beer box and more than your garlic work as well. I'm assumeing your useing a proper cutting board and the beer is the typical 12 pack boxes. To remove the burr take a stroke or two per side at a higher angle as your very last sharpening strokes and see if you notice any improvement.
I'm pretty sure I'm getting rid of the burr. I feel for it as I sharpen and when it turns to one side, I do strokes on that side trying to eliminate it. If I happen to flip it to the other side, I do strokes on that side and repeat as necessary until I can't feel it any more. I also occasionally run the edge through the end of a piece of cardboard, lightly, just like many people do with soft wood endgrain to remove the burr. I generally follow that step up with a few alternating strokes, and again, test for the burr when I'm done.
As for my cutting: I just resharpened my chef's knife and butchered up a six pack container. Not the heavier 12 pack containers that are corrugated: The lighter single layer stuff that six pack holders are made of. I did around 15 cuts until it was in smallish pieces; half and half push cuts and slicing.
When I was done, I felt the edge and it had rolled slightly to one side. It still mostly popped hair, cutting both directions though. 3 strokes on the strop on that side, and it was rolled slightly the other way. A dozen more back and forth, feeling for it, and the edge was aligned again. My previous post was from memory; this one is accurate.
However, when I sharpened this time, I decided to stop at the Spyderco medium ceramic (600 to 800 grit), and then strop a bit on CrO loaded leather. I'm not sure if this helped, or was the same as my previous experiments.
Finally, yes, I'm cutting on a proper cutting board: Made of bamboo in this case. You have to understand that when I say I chopped garlic, it wasn't a 30 second operation. It's more like 4 to 6 minutes of nearly continuous chopping, piling it up, and chopping some more, plus all of the stemming and slicing before the chopping. I'm sure my knife technique could use some help, and I'm also sure that the rocking of the knife on the rounded part of the blade is hard on the edge.
I really should have started a thread about this a while back. Maybe I will later if I have the motivation.
Brian.