There is a common mistake to see aikido as a fighting art. It is an art based on the first reaction and a dodging skill is essential. I think, that the idea of relaxation is fine, but definitely in dojo. In daily life it depends on our assimilation to an environment and our preparation to deal with obstacles. For example, it is visible who is prepared to take an exam (any), and who is not. Our brain decides how to choose a self-defense reaction. Any extra breathing, a posture, actually do not help.
Perhaps we have to agree to disagree
I didn't say only
relaxation. I also didn't mention extra breathing, although breathing is very important. A relaxed posture with appropriate focus in tanden is I believe a good foundation in allowing better reactions to events.
As to the brain deciding things, I am not so sure. For fastest reactions the brain does not have time to decide things - responses need to be instinctive. This can be trained, although it is not easy.
I don't think the definition of aikido is that simple. You are picking on a single event to define aikido. I think there are many other events which also contribute. Ultimately, I am not that bothered by your precise definition, I am working to define it for myself. I wish you all the best for your personal journey of definition!