My take on part of this issue is that what Tohei has shown quite well, is that the basic level(s) of what I call Internal Aiki, and the power that is one result of it (or, one level of it's application), can be learned and accessed through internal awareness and feeling/kinesthetic training, using relatively simple and straight-forward methodologies.
The thing is, there is a lot more that needs to be learned beyond that in order to externalize it properly, and actually apply it in Aikido. But without even the basic knowledge and training to make these internal connections, Aikido becomes, as has been said, "simply a form of Jujitsu" (not a bad thing, but in actuality very different.) This is mostly what I have seen in the Aikido world over the last three decades. If one wants to "bring back Aiki to Aikido", Tohei's "Ki training and mind/body coordination/unification" is a very reasonable place to start.
To me, though, it has it's limitations. Unfortunately, (forgive me my Ki Society friends) even though Ki Society training in this fundamental area is good, in my opinion it isn't always applied, or applied properly, interactively to actual Aikido practice and execution, and that's where things start to fall apart, so to speak. (I have known some very good older/upper level practitioners though.) Also, I think Ki Society has become over-identified with their teachings and training methodologies around Ki and leading the mind, such that some other things tend to be neglected. (Similarly, to me, many Aikido styles tend to be over-identified with external technique and neglect, are mistaken about, or are devoid of, the internal training, thus also being out of balance and "incomplete.") Coupled with the fact that, for various reasons, Ki Society training tends to be fairly isolated from other organizational training, I think this is why people would see Tohei's "attempt" at bringing this fundamental aspect to Aikido, and the expected results, as unsuccessful. In a real sense, it's true.
To me, what it ultimately means to both learn and then apply "Aiki", at different internal and external levels, to Aikido practice and actual execution, is the tricky and interesting thing. There are other methodologies than Tohei's available, though not many. To learn it, and learn to access it, it is best to be inducted into the experience by someone who is well-versed in it. It's difficult to find on one's own without being able to feel it. But to me, this stuff comprises the core elements and very heart of Aikido.