I am on the fence about this one, mostly because [I think] the way this is explained by seniors and the way it is interpreted by juniors is sometimes not the same. This is one of those concepts that is generally introduced too early and the result is... different than the intent.
The goal should be to throw until you know how to throw and can do it with competency. Dan is spot on in saying throwing someone who is not cooperating is a different experience than throwing someone who is actively participating in their own compromise. For those who can competently throw tori with or without tori's participation, talk all you want about the mundane of focusing on throwing our partners. As a competent practitioner, ideally your movement transitions such that you do not focus on moving your partner; you focus on moving yourself after assimilating your partner onto your center.
I'd also add a few observations:
1. Aikido is about control. The fact that you are trying to alter your partner's response to your own design inherently defines your actions as "controlling". That said, I do not think there is anything wrong asserting control over your partner; the magic lies in why you are asserting control...
2. Aikido in many respects seeks to accelerate the experience of throwing by generally repressing uke's natural responses to preserve his center. I think we call those who do not willfully fall for us "jerks with bad energy"...
My instructor used to describe this transition as analogous to when we learned to drive. Both hands on the wheel, full stops, no radio, turn signals when we think we are changing lanes and absolute terror merging onto the highway. 10 years later... listening to the radio, driving with one hand while (gulp) texting and absolute terror merging onto the highway. We need to let our bodies absorb the movements before we are ready to decide what we do and don't need. And we certainly should be questioning anyone who has decided to tell us what we don't need or won't teach...
I know it's the norm to think Aikido is about control and that may be so for a while but in truth it's about quite the opposite of such. A very hard concept to understand and as I observe even those here 'certain' of their 'expertise' still hold on to this concept of control.
Ueshiba actually pointed this out but others find that hard to believe. When you watch someone and the ukes are flying about, or crumpling on the mat then it seems 'obvious' they are being controlled to the observer and the observer then wants to learn how to do that.
However, when he is told by the 'master' that he himself is not trying to control at all then the observer would find that hard to accept.
Anyway, that was just to let you know so that if you ever feel like that in Aikido, that you weren't 'trying to do anything' then you will remember what has been said and not get too confused.
Being somewhat 'controversial' in my views when comparing them to the 'norm' I will add something on the second point, throwing. Look at it as receiving, taking, projecting. This I would say is the first differentiation needed rather than the lazy use of the word throw.
Now, center? Nothing to do once again with anyone extending center or not really. If we use center as the point to focus on regarding projections then it would be connecting, receiving, taking and projecting. As you can see from this breakdown that takes quite an amount of skill.
Add to that the fact that you could take someones center and project, direct them straight into the wall.....smash!!! (or pavement for that matter) but then to see why that wouldn't be Aikido for Aikido is Harmony. Thus the practiced of projecting them into a circle, for their own safety....harmony.
It's a useful exercise, just like learning the sword is a useful exercise and both take great discipline and study and practice. Therefore it boils down to have you learned to do it yet? or can you do it? That's all really. Not whether it should or shouldn't be done.
Then of course there is doing the same thing, projections, where the focus is not on center and for instance on leading or even taking and leading the mind. Just as effective, looks exactly the same, however, different skill. Different study.