It would be a bit like changing the Qwerty keyboard - you wouldn't start with it (how about an 'Abcdef' board) but you are stuck with it.
One solution might be to use IATA codes but to add numbers to increase the number of options. This would give 34* ^ 3 = 39,304 options. *exclude 0 & 1.
It is of course easier for most US airports where you add a K (For example Key West is EYW & KEYW). I don't know if you could extend it worldwide so that you added 'E' to any European airport to get the ICAO code - you would have to change all the ICAO codes (see my first point).
Someone told me that in the early nineties ATC switched from IATA to ICAO codes overnight - is this true and if so how did it go?
I have to say that I am pleased that I can type an ICAO code into google - its helped me many a time when browsing PPrune.