Since I don't have mad skills, I used an alternate method which I will refer to as the "Screw Jack Method." If you've ever used a screw jack you know what I'm talking about.
I had the screw out and had the technique down. now there was only another 7 plugs and screws to remove.
I swapped out my deck screw for a much beefier wood screw that I have no idea why I own. I have a box foll of these. I think there are something like 100 of them. They are 3" long and about as big around as my pinky. Since I was hitting a solid bed of epoxy after about 1/4 inch, the screw diameter didn't matter as much as how aggressively the taper bit into the wood. With the larger screws I had much better success. I still found that the worst screws though were the ones at the ends of the rails.