I actually felt I could offer something here. From my understanding - having trained in a few styles of aikido, and teaching at a Nishio Aikido-based dojo - the study of the application of atemi is fundamental for learning and understanding body positioning.
Atemi does not have to be used at all during a technique. But... if we don't know where it's applied, then we have no realistic reference for body positioning and angle relative to uke.
If uke attacks, and can strike, but nage can't strike; then nage's not in a very good position to do anything else. In other words; nage is dead.
Atemi acts as a compass, ruler, slide rule, etc. Just as the sword acts as a tool for refining movements - especially concerning the angle of the hands at any given moment in the technique.
A sword in hand is not at all necessary in order to apply effective aikido technique. But a sword is vital for learning how to apply effective aikido techniques. Flowing techniques can absolutely be executed effectively with no atemi applied or any where in sight, but a knowledge of atemi is necessary in order for the flowing techniques to be martially effective.
Someone not training atemi in aikido will not understand the angles and distances of nage and uke. Learning atemi application is like learning the ABCs on the long road of learning how to "write" in aikido. O-Sensei's movements in his later life were still absolutely martially effective, even though it may look to many as if he's just waving his hands in the air. And that type of more flowing aikido could be likened to someone writing their signature. But in order to learn to write a flowing signature, we all had to start out when we were kids and learn how to make letters, then spell words, then write sentences...
My signature shows little remaining evidence of my sitting in grade school learning how to make block letters, and then later, cursive letters. But it's all there.
Atemi is like the scaffolding erected to construct a building: it will serve as a load-bearer, and a frame for proper angles and measurements. And, of course, after the building is constructed the scaffolding is not necessary for the building to stand.
And the role of uke is equally as important in the study of budo as the role of nage. If nage is not applying atemi, uke will not learn about the strike possibilities at any given moment within the technique. As an example, if uke is put in the position to have kaiten nage applied, among other things, they need to know that nage is in the position to be able to plow a knee right into uke's face. An uke trained in atemi will naturally put their hand up to their own face in the direction of nage's inside knee.
Any aikidoka trained in atemi application can easliy spot someone else in training who doesn't have the knowledge of atemi. They're easy to spot because they're often in the position of being wide open to incoming force/s from uke. And I see this even in some high-level aikido practioners. They're open. Which equals: they're dead. I've seen many people training aikido focus so much on the controlling technique being practiced - kote gaeshi, shiho nage, nikkyo, etc - that comes near the end of the technique; that they blindly dance through the initial opening movements not seeing that they're completely open to attack. And those who do not train atemi will also be blind to attack possibilities from uke even during the controlling techniques.
In my training with Nishio sensei, he seemed to have little concern for which model of the controlling technique people chose. To him they were even more of a "massage" for uke. He always stressed being in proper position from the word go to strike uke (in many cases multiple times) and to not allow uke to be in a position to strike. And to continue that relationship throughout all the movements.
Atemi is literally the "inside scoop" to aikido as budo. There is an old masonic creed: measure twice, cut once. And one of the symbols of masonry is the square and the compass.
Atemi and the sword are the squares within aikido. Atemi, in unarmed applications, is the sword of aikido.