All the groups ultimately want to improve the world and the people training in their Aikido. Everyone claims martial efficacy as a goal. Takemuso Aiki, the spontaneous creation of Mushin technique, is something pretty well every branch of Aikido, if not every martial art in the world holds up as a goal.
The question is How.
Our English language has 26 letters, 5-7 vowels making 20 separate vowel sounds with a total of 44 sounds, and a pile of grammar rules that many experienced English-only communicators frequently mangle. Our written language is represented in calligraphy, cursive, printing, and an ever increasing number of font. We have jargon for specific uses and groups. We have dictionaries and thesauruses (thesaurasii?). And, we allow poetic/artistic license anytime any of the rules don't serve us. But, we start written English with 26 things that can expand eventually to Dick and Jane, Ulysses, a Brief History of Time, and the Bible. 26 things that give us unlimited potential.
When we learn mathematics, we learn the language of the universe. Physics bears witness to all creation. The greatest mathematicians still started with 0-9, then learned how to manipulate them. Ten digits that can be used to predict the weather, the seasons, the depth of the ocean, the power of the atom and the location of interstellar bodies.
The most basic and most powerful computer programs all are derived from 0 and 1. Like the Yi Jing, Wuji (Nothing) gives birth to Taiji (two things - something and the absence of something), the Four Symbols, the Eight Trigrams, the 64 Hexagrams and so on all the way up to limitless. Whether it is Yin and Yang, or Binary machine code, two things can eventually lead to potential we have yet to fully realize.
The Founder and his teacher apparently didn't build on basics so much as dropped the equivalent of astrophysics on their students and left them to figure out how to frame their own insights and communicate to their students in turn. Maybe it's not surprising so many people developed different ideas.