So, your question is gradually developing to debating
The idea you are promoting would be excellent in an ideal world. But the world is not ideal
On the one hand, you are right: everything CAN be explained (at least, in theory).
On the other hand, as I told already, very often native speakers do not know an explanation. You have to know how the language developed in its history in order to understand anything logically. And that gives many times more information for you than just to learn its modern state. You can learn the modern state of a language more or less easily, but you will never be able to understand why it is this particular way and not any other way, unless you study the historical linguistics.
Finally, different languages are different ways of thinking. What seems to be logical for a native speapker, does not seem logical for a foreigner.
You know, my wife is currently learning English. She is at the very beginning level yet. Recently she has learned the usage of "OF", and then she made some grammar exercises.
And she asked me a question very "symmetrical" to what you are asking:
I understand that I need "of" in the phrase like "in the shade of green trees" (в тени зелёных деревьев), since it is genitive in the Russian phrase (деревьев).
But why shouldn't I use the "of" in the phrase like "five tall green trees" (пять высоких зелёных деревьев)? It is the same genitive there! Don't I need "of" in English?
What is nonsense - is to ask about the grammar case of an adverb. Adverbs do not have cases.
Now some people say "много" is not really an adverb. But it is a question of terminology. No matter what we call it, it behaves as an adverb.
Now, what you are saying (хороший – хорошо, трудный – трудно) is correct! Many adjective have related adverbs. The same as многий - много. Very similar!
But you can talk about the case of an adjective (хороший, хорошего, хорошему, хорошим, хорошем) and you cannot talk about the case of an adverb. It makes no sense to ask "what case is хорошо". I hope you get my point.
Impulse, I am sorry, but I did not understand your example with "a lot of". Your explanation does not make much sense to me. I just know that I have to say "a lot OF cars" but "many cars" without "of". The same way as I remember I have to say "hundreds OF cars" but "one hundred cars". It does not seem logical to me, but I've got used to it nonetheless.