Good for you that you've tried to make rhyme and reason out of the general principles Tohei imparted.
<passive-agressive-sarcasm>Firstly, thank you for taking the time to patronise me on the internet. I enjoy being spoken to by strangers in the manner that I congratulate my 3 year old son when he successfully uses the toilet. Makes me glad to be here on aikiweb.</passive-agressive-sarcasm>
I'll just add a smiley so that humour is indicated, thereby attempting to gloss over my unpleasant statement, it's one of the things that always works well on internet forums
How has this work tied to what you've experienced from people who've trained with Dan and Mike, specific to spiraling / silk-reeling? This is the specific portion of the set that Tohei's subset does not address, unless it's some kind of okuden.
And, no worries if spiraling falls into the category of "being a good addition", and being one of those "missing" things. One of Dan's key admonitions is that practitioners focus on spiraling and other more complex movements -- over foundational connection and opening skills like one-point, keeping weight underside, and refining the ability to extend ki -- at their own peril.
I studied chen style tai chi for a number of years just before I began aikido and stopped shortly after I found aikido. I went back to it for a little bit not long after I got my shodan and found it complimented my training nicely, but I moved away and never took it up again as there was no teacher where I moved to. Though a teacher has moved near to where I live now so I've been thinking about starting again. All in all I'd say I got about 2+ years of fairly regular training in it alongside aikido, not much but enough to be useful to me in conjunction with my aikido.
One of the interesting things about spiraling and silk reeling as I understand them (and keep in mind I've not practised them with a teacher for some years here so could well be wrong) is that of course you're quite right they aren't really in any of Tohei Sensei's standard ki tests, nor can I remember them ever being actively taught to me. But if you look at some of Tohei Sensei's waza, particularly things like kirikaeshi which you see done a lot in taigis it's right there to some degree. I'm basing this on the fact that learning about some of those things helped me figure out how to do the aikido properly. So it could be that I'm cutting and pasting them into aikido unintentionally and they were never there, I just added them of my own accord, if that were the case then I'd say we're probably in complete agreement about all this, though I think there's probably still room for debate. If the only way for me to do those waza correctly was to bring my tai chi experience to them was it because the tai chi was the missing piece? Or was it because it was there all along and I'd simply not figured it out or been actively taught it yet? I don't know, all I know is that my teacher at the time commented that I was getting better at it and that he could do it just fine despite never to the best of my knowledge having studied tai chi of any flavour.
In the sayu exercises of Tohei you can see at the 8.17min mark here http://www.youtube.com/watch?v=XuuhW9t11e0
I found silk reeling to be very helpful as I recall, though we're going back a decade or so and my recollection is a little fuzzy. Note that what you see on the video isn't the exact exercise I'm thinking of so it may strike you as me talking nonsense, it is however, the only video of Tohei Sensei I could find where he's doing anything like the exercise I'm thinking of, nonetheless I hope it should be reasonably apparent that silk reeling could be quite helpful in this aikido waza in certain contexts/variations. Sayu as shown in the video is one of the most misapplied technicques I've come across in the aikido canon. I've seen people ranked as high as 6th dan doing it by basically getting uke close to them and then just smashing them across the face with their forearm/elbow. That's just ugly. I've also heard them describe it as being the Daito Ryu way of doing it and that they obviously come from a more 'traditional' style of aikido than me, never mind the fact that what they're doing is most certainly *not* the Daito Ryu way of doing it according to my limited exposure to the art (outside of aikido). Suffice to say my understanding of the technique is that when done correctly there need be no smashing or crunching, uke simply cannot remain standing because nage's outstretched arm is in the way, this arm should not yield if pushed or pulled, uke's only escape is to either move backwards, which often happens if this is done statically, or if done dynamically their forward momentum means they must drop their head and upper body out of the way as they travel through.
Regarding spiralling, and again you'll have to forgive my rustiness if I get this wrong, it may interest you to know that within the style of aikido I now practise the founder - Koretoshi Maruyama - has introduced some exercises that bear a striking resemblance to spiralling. He observed that after a few decades many people in the Ki Society were getting bad backs, he attributes this to them holding themselves in a certain posture whilst always keeping one point. Consequently he now emphasises focusing on the sacrum which he says amounts to the same thing as keeping one point. He also has exercises where you imagine a ball in the bowl of your pelvis which you roll around, he encourages this in warm ups so that stiff backs do not result after long years of training. It's worth noting that he doesn't link this specific exercise to hand movements the way it might be in tai chi (well if he has I've never seen it done), but it's intriguing nonetheless. Rolling imaginary balls around in your hara is not the only time I've heard him speak about imaginary spheres (I can't bring myself to use the phrase 'big ball of ki' for some reason), I've heard it in other contexts, I'm reasonably sure it's a Daito Ryu influence coming out in his teaching, I don't think he spontaneously created it himself. How these things came to be in his technical repertoire is another issue entirely.
PS - sorry about the sarcasm earlier in the post, perhaps you see now why I left aikiweb for many years...