If I had sufficient skill, here's how I would teach aikido. All comments are welcome.

Stage 1: Ki and kokyu ryoku
Goal: Develop and condition internal body skills
Means: aiki taiso and breathing exercises
Explanantion: Think Mike Sigman, Dan Harden, Rob John, Akuzawa.

Stage 2: Applying internal body skills
Goal: Applying the skills of Stage 1
Means: ki tests
Explanantion: Stuff like grounding a push, push/pull someone with the rowing exercise, tenkan from a wrist grab, etc.

Stage 3: Aikido kata
Goal: Learning the martial/aikido principles inherent in Stage 2
Means: Basic aikido curriculum
Explanantion:
This is not about techniques! Aihanmi katate dori irimi nage is not a technique. The katate dori is the starting position. The irimi nage is a throwing principle: unbalance partner to the rear and cut him down. Ironically, if you fail to unbalance your partner at first contact (as learned in the Stage 2) there is little use for the actual irimi nage...

Stage 4: Aikido waza
Goal: Learning the applications of the principles/ideas
Means: More dynamic and free practice of the basic aikido curriculum
Explanantion:
Use your understanding of the principles of Stage 3 to create techniques in a more dynamic environment. Irimi nage may not look like the standard irimi nage, but the principle needs to be present. This Stage still uses the uke-tori type of interaction.

Stage 5: Randori
Goal: Learning how to use aikido waza in a dynamic, alive, resisting environment
Means: Let people play around with aikido waza (think à la pushing hands)
Explanantion:
Two people playing around, trying to apply Aikido waza on each other.

Stage 6: Kumite
Goal: Learn how to fighting
Means: Sparring, free fighting
Explanantion:
If you want to learn how to fight, then practice fighting. Basically, anything goes here, as long as the contents of Stages1-5 are present. And you're still friends afterwards.

Some more remarks:
You start out at Stage 1 and Stage 2. As long as you have not developped some basic internal body skills, progressing to Stage 3 or further will only destroy the progress you made in the previous Stages. This applies to all stages: the later ones build on the previous ones. But it is also true that later stages will shed new light on previous ones.

Students should do the Stage 1 exercises at home. In this regard Aikido is like learning to play a musical instrument. If you only practice while in class, you will not make significant progress.

Disclaimer:
As I was writing this down, I noticed some resemblance to http://www.aunkai.net/eng/bujyutu/index.html. Other influences are present as well.