Something that happens all too frequently when not using a countersink bit is described in detail, although the effort when taking into account all the screws you are going to put in this machine was WAY glossed over, and no one else commented on, was stated here:
I got about the same results, though without the coutersink hole, it did "frey" the edges a bit. My solution for that was to put the screw partially in, cut off the frey with a utility knife blade, and then finish screwing it in and sink the head. I puttied over the screw head and it all was good. Saved a lot of time vs. using a countersink bit.
Now, color me ignorant here, but screwing in the screw partially, cutting off the fr
ayed crap that will happen, and not gouging out crap you DON'T need to be taking off, and then completing the screwing in of the thing seems to be more work than using a countersink bit.
Part of that "ignorance" probably comes from thinking you are using the right tool for the job. I'll provide you with a picture for this, as it seems you spent money on a tool that works fine, but if you had spent a few more bucks, would have gotten you the tool that is SPECIFIC for your use, and PERFECT in its job of saving you time and giving you a neater appearance.
The thing you want it this here:

If that one's too expensive, Home Depot will sell one of several quick-change countersink/screwdriver sets, and all you have to do is buy the screw tip you need and swap it out if the standard #2 phillips bit isn't what you want. They'll sell either a DeWalt, Ryobi, Bosch, or Milwaukee brand, or perhaps all of 'em at once. This thing is EXACTLY what you want, and will eliminate you sittingthere with a friggen utility knife cutting wood by hand, when you should be doing it with *insert Tim "The Toolman" Taylor voice* - MORE POWER!