My first instructor, the late Hamish Macfarlane, told me that aikido is 90% mental. This does not mean that the physical aspects should be neglected, but that they are not all there is.
Centering is particularly important in aikido. Centering is physical (turning about your center, extending from your center), mental (relating all actions to your center), and emotional/spiritual. The last item is what I want to address.
Self defense is a scary and emotionally difficult situation. When somebody physically pushes, you turn. When threatened, you do the same thing: you don’t allow yourself to be moved emotionally, but rather you blend and turn.
When pushed physically, you retain a strong center, but do not use it to push back. You can do this, of course, and it will work if you are the bigger stronger, person. But that is not aikido. When you turn, you blend with the incoming energy, you add your energy to the incoming energy, and steer it to where you want it to go, typically around your center, then either let it go, or direct it into your technique.
When pushed emotionally, spiritually, it is equally important to be resolute and immovable, while not using this to resist here either. Rather, blend and turn. Even if you are logically correct, saying “you are wrong” is going to cause conflict. If, instead, you say, something like, “that is an interesting position”, you are making contact, forging a connection, just like when you avoid a physical attack and make physical contact with the attacker’s center. Then if you say, “can you tell me why you think that”, you are adding your energy to the incoming energy. Then, you can ask questions to lead the energy of the discussion, perhaps to where you want it to go.
Once you have control of the situation, physically or emotionally, what you do with it is also related to aikido. You can in either case cause a lot of damage, but the way of aiki is to minimize the damage to an attacker. Physically, we use the energy to throw uke into an ukemi that is both survivable and within their capabilities. We can do the same in an emotional conflict.
In either physical or emotional aikido, practice is, of course important. Knowing how to do kote gaeshi does not mean you can actually do it. There are subtle movements of the body, arms, legs, etc that have a huge effect on the effectiveness of the technique. The same applies to non-physical aikido also.
We do typically practice the physical sort of aikido quite a bit, a lot for some of us. We probably practice the other aspects a lot less. I was never taught these other aspects, so I probably don’t address them enough either. I was expected to learn them through osmosis. Certainly the various aspects of aikido are not separate. Studying the non-physical aspects will also enhance the physical ones.
Finally, just as different practitioners have different styles of physical aikido, there are different styles of non-physical aikido also.