Logan, I'm always amazed at these kinds of questions because they were all addressed in the training system I came up in. When I visit other aikido schools, I see none of it, so I'm not surprised that people wonder how to deal with complex, aggressive attacks and others say you have to go outside aikido technique to handle it. Aikido is swordsmanship. Would anyone tell a swordsman, "You have to go outside swordsmanship to handle this situation?"
If aikido is learned as a martial art, there is a definite progression to the training that takes care of all your concerns.
Begin with the various grabs and learn the basic techniques.
Add punches and develop dexterity applying the techniques against various strikes.
Add kicks. (Note that this means that both partners, by now, are proficient at punching, striking and kicking, as well as all aikido waza.)
Add randori. First, shite randori, in which every technique is specified. Then go to jiyu randori, in which attacks and defenses are not specified. At this point, attackers should come from every direction and with every kind of attack and weapon. Attacks should be powerful and serious, but controlled. People trained in striking and kicking can stop their attacks without hitting the defender, though demonstrating that their attack can penetrate. At this level, aikido technique must be very clean and to-the-point: aikido is swordsmanship. When the attack comes, you cut it down. Nothing more, nothing less. Of course, you should also have the control in aikido to bring any attack to a safe conclusion, meaning that the attacker's own movement is drawn irresistibly into a position where the attacker cannot move at all, cannot generate strength against the defender, is in some reasonable amount of pain, but is not injured. The aikido defender has total control.
Next, up the randori: jiyu chikara randori--free attacks and aikido applications with both partners using strength and resistance where applicable. If the attacker is not thrown in the instant of his attack, he doesn't throw himself but attempts a follow-through attack. In the yoseikan, this follow-up attack could be a second punch or strike, a footsweep, any kind of throw or grab, and it will likely lead to a grappling situation on the ground. There's plenty of aikido in grappling, too. It doesn't end when your feet are off the floor or when your back is on the floor. And in grappling, the original defender either prevails or "gets killed." Every encounter is a serious study of life and death. Exhaustion is a great teacher. Eventually, one learns to take every attacker off his feet and into a locked-down position in the first instant.
Now, if your school doesn't teach this way, it's because the teacher wasn't taught that way and the school can never teach you to reach that kind of level. But if you learn from someone who learned that way and teaches that way, you can apply aikido technique to any attack without going outside aikido technique--which is formless, anyway, so how can it be limited?
The only way to answer your question is through research in training with people who can do it. And that means, you have to ask yourself, how important is this to me? It might mean you have to give up your life and comforts and go somewhere where you can really learn it. And anyone who does go to a high level will naturally take any opportunity to learn anything else that can bring him or her greater understanding.
Aikido is a sword, but it must be sharpened against a stone. And that stone can be any of the other martial arts, such as karate, judo, jujutsu, kendo and kenjutsu. The best aikido has been sharpened with every stone.
Budo is great in that it allows us to experience losing and death, time after time, which gives us respect. And it allows us to work diligently on preserving life.
Best wishes in your search.