One thing I forgot to mention regarding the whole 1+2=3 thing is not only the fact that focusing on 3 (kake) without paying much attention to 1 (kuzushi) and 2 (tsukuri) will often not pan out, but focusing on the throw will often keep us from being able to flow from one technique to the next. Many times, whether with aikido or judo, the first shot doesn't get him. Sometimes, it's not even the second shot. But by the third, things really start to fall apart for uke.
One of the most common rookie mistakes I see is A) going right for the kake, and then, when it fails B) going back to "neutral" or square 1.
Get 1, then 2 and see if 3 happens, if it doesn't, don't just hit "reset" and start all over; step immediately to the next kuzushi and tsukuri and see if the next throw happens, etc.
I like to think of it as putting the man on the edge of a cliff. He'll teeter, and either A) fall off the cliff, or B) do something to regain his balance and lurch back onto solid ground. Take advantage of that recovery action, and put him on the edge of another cliff. Eventually, he'll do a Wile E. Coyote and zip over the edge before he knows what's happening.