TL;DR - Martial arts organizations are generally a form of quality control that will rarely, if ever, interfere with your learning experience outside of a few specific occasions such as grading for higher ranks and establishing competitive rules.
I'd like to start with the last part of your question :
Wouldn't it be perfectly valid to be able to practice martial arts without such organizations? And still be "licensed"?
Yes, it is. And for the most part, that's (paradoxically) exactly what training under such an organization should look like.
No one is forcing you to join such an organization. Unaffiliated schools are everywhere. Some of them even have legitimately skilled instructors! But that's the thing, really. Unless the instructor in question is famous for being a former champion or something, how do you know if he's actually good? And even then, how do you know if he's a good instructor? Sure, you can go to his school and take a few trial classes. Maybe he's a great teacher, but maybe he also doesn't know much about martial arts. And if you don't either, how will you tell?
I guess that's my main point. Generally speaking, martial arts organization will keep public records of their members, or at least their higher ranked ones. So if you see a school that's affiliated with, for example, the International Traditional Karate Federation (ITKF), you can contact your local ITKF chapter and ask them about this school. They should be able to tell you if the school is indeed affiliated with them, as well as give you minimal information about the head instructor there (like his officially recognized rank and when he was promoted to black belt, for instance). You can then compare this information with the instructor's claims. That way, you have some assurance that your instructor knows martial arts, and now you can focus on the teaching aspect during your trial classes.
And that's basically what being "licensed" means: there is someone out there willing to vouch for your qualifications and that will recognize your rank, as well as those of your students. So if you ever move away from your school, you should be able to integrate pretty seamlessly in any other school affiliated to the same organization.
Unless your school is near the association's headquarters (or "worse", is the flagship dojo), the odds of someone from the association actually dropping in and taking over the class are pretty much inexistant. In fact, if someone actually drops in like that, it's likely to be planned long in advance, with your instructor inviting a higher ranked guest instructor for a few classes or seminars. Outside of that, the only time you're likely to even hear about such an organization is when you are up for a promotion at a higher rank (such as black belt or higher). Because that's how you get your own "license", and the association will want to maintain a certain quality level and award these higher ranks themselves.
My old karate school would actually only require you to officially become a member of it's parent organization when you reached ikkyu, as you were required to be a registered ikkyu for at least one year before becoming eligible for the shodan exam. Even then, outside of the examination committee, you'd never actually interact with the organization. In essence, it was functionally identical to being unaffiliated until you were ready for your black belt exam. Also saved you a few dollars per year in membership fees.
Now... do these conditions affect your training at your local gym/club/dojo? they shouldn't. Your local instructor is the one planning the lessons, he's the one doing the teaching and he's the one awarding ranks (up to a certain point). He might have to teach a specific curriculum for each rank, but that's a normal part of pretty much every education system known to mankind. He might have a few assistant instructors coming from the higher ranked students, but that's often a necessity with larger classes.
Instructor, assistant instructors and students, that's the extent of the hierarchy you will personally deal with, at least until you reach much higher levels or if you compete in the parent organization's events.