The bike goes where your eyes are looking. I have no idea why this works, but it is amazing when applied.
The key is to always be looking as far ahead through the corner as possible, when you start turning in you should be looking at the apex, once you have reached the apex you should already be looking past the exit. I've passed cars and motorcycles and taken many downhill KOMs using this technique, while on 23c tires!
If you watch any motorcycle race or go to any motorcycle class/school, their eyes are always ahead of the curve their on. This takes a lot practice to get down especially when your cornering confidence isn't very high.
I have a very similar cornering technique to what you described and I have the same problem you described, where I can make it to the apex just fine, but need to bail out when I get there, this is because I haven't properly sighted through the corner, almost every time. (I watched someone crash in front of me just last week doing this, they saw gravel on the outside of the apex and went right for it, instead of looking at the exit)
I find that slick tires (assuming you're cornering on asphalt) offer much more surface area and therefore grip than semi knobs. The knobs decrease rubber contact with the ground and tend to deform when cornering. (I learned this both the hard way and easy way).
I prefer to decrease my tire pressure when cornering in order allow the tire to deform to the road surface and therefore have more contact with the ground. I use this to determine tire pressure. This also allows more absorption of small bumps. A very hard tire will bounce over everything and when you're bouncing, you are no longer in contact with the road surface.
You can also utilize different lines through a corner in order to achieve a different exit through the corner. The fastest line through a symmetrical corner will always be a symmetrical line, but it won't always be the safest. A late apex works great also
As someone quoted in the "rules," practice is the best way to increase cornering confidence, well...and maybe a little bit of fearlessness.