I don't understand what do you mean by 'having structure' or 'not having structure'. I find it very weird wording. Universally understood word in aikido is rather 'position' 'posture' or 'alignment' of the body. If this what you mean, so I disagree completely. In Hand 2 Hand situation both ppl can have correct alignment. It is easy to see i.e. in boxing.
Structure, or integrity....posture or alignment can work too I suppose, although, you can have structure or integrity and NOT have "posture" or alignment". The semantics of those two words I think don't really capture the concept as you can be odd positions.
Sure you can reach a situation in which both people have integrity and are simply "trading blows" in most H2H situations though it is typically short lived with one person getting the advantage over the other.
Of course it is easy to see in boxing...you have a ref that breaks up the fight and re-establishes the parity in the situation once it is lost by one of the fighters! Actually this is a VERY good example of what I am talking about and is really the big difference as to why MMA is vastly different (for example) than boxing. In MMA the fight is continued after structure is lost and allowed to continue until one of the opponents is no longer able to fight back, the fight stalls to a degree of bordom, or the situation becomes too dangerous in the eyes of the ref and a winner is declared.
Of course we can practice this in Aikido as well and I am sure most do to some degree. Kaeshi waza or reversals is how we do this. Uke attacks and give up structure/integrity and then works to regain it. It can involve timing, as timing is always present in any situation, however, I think timing is a minor element as to me timing implies that I am "waiting" to find a gap in nage's technique and exploit it. So to me timing is EXTERNAL to YOU and depends on nage. What is more primary is what is going on INSIDE of you and what you do with your OWN body to regain your integrity and reverse the situation. You will, at least in theory, cross through a postion where both uke and nage have reached parity, in most cases I believe it is a brief point, however sometimes it is not and it manifest as a "struggle". This is the point that most of us concentrate on reducing or avoiding as we want to be on the side of "winning" which means simply that you have broken the other guy's structure/integrity.
So when we get in that "struggle" we need something that is going to turn the tables and disrupts his "system" allowing you to gain control. and that is the essence of what we are talking about I believe is at that point in time where you need to break his structure. It can be atemi, a weapon...it can involve timing, a distraction, or you can disrupt his feedback processes through your own body, which involves various methods of reducing tension, proprioception or "shifting" etc.
Specifically, this is what will in reality, I believe, give us a distinct advantage over our opponent when all else is equal (or not equal). "all else" meaning speed, timing, weight, strength, and technical skills...things like that.
Hope this helps clarify my perspective on things!