The pic above is a perfect illustration as to why you can't simply 'hide' a headset on a frame designed for a standard, external headset. If you look closely at the headtubes for both the integrated and internal options, you'll see the slight flare where the headset bearings are located. Without this enlarged diameter of the headtube, there would simply be no available space for a proper bearing. You can do the math if that helps make it even more clear: 25.4mm diameter fork in a 30.2mm inner diameter headtube leaves 2.4mm of space between the two parts. The cartridge bearings for an innicycle are 5mm in diameter, and that's before you include the space for the inner and outer race. Maybe you could squeeze 1mm diameter bearing balls between the fork and headtube to hide the headset? Maybe it would last a ride or two?

The cycling industry has done some funny things along those lines with bottom brackets in particular. If anyone remembers the ISIS 'standard' it is a great example of pushing things too far with regards to minimum bearing sizes in an attempt to increase the crank spindle diameter. There are other more recent examples with attempts to convert from some poorly chosen integrated bottom brackets back to a threaded setup but (Trek BB92 I believe). Anyway, it's not a path I'm interested in pursuing for those and other reasons.