Using Forstner bits instead of hole saws
I'm not sure why someone told you that you can't drill all the way though with a Forstner bit, but in my opinion, they are incorrect.  These bits excel at creating a flat-bottomed hole but will also drill completely through the material just fine.  I regularly use mine in a drill press but they can also be used in a handheld drill.

I find that they splinter the material much less than a spade-type bit does although they are significantly more expensive.

Here is what Rockler says about them:

Rockler


  What's so special about Forstner bits? 
  
  Unlike the standard twist bit, Forstner bits are optimized for
  woodworking applications and cut exceptionally precise, clean-edged
  holes in wood. Because they're designed to produce a minimum amount of
  tear-out when exiting the material, Forstner bits are the best bit for
  drilling through holes. Forstner bits drill a flat-bottomed hole,
  making them a necessary tool for many hardware installations where a
  precise depth of mortise is required. Forstner bits are guided by the
  wide outside rim of the bit, unlike most drill bits, which are guided
  by the tip. Because of that, they can be used along with a drill press
  to drill angled holes, holes that partially overlap, and holes on the
  edge of the material.