Vocabulary or Grammar? (part 1)

Which one is more important? Vocab or Grammar?
Definitely both. But how should we see these two aspects of language for us to better understand their roles in communication?
Let me give this illustration (I read this somewhere; this is not originally mine). Communicating is like delivering a package of boxes. The knowledge allows you to put one box before another. But what you put in these boxes - the message- requires vocabulary knowledge.
What use are the boxes without the content?

Our education system so far, or at least what I experienced in my school years, has been focusing too much on the learning of rules. What should come at the beginning, and what should follow, this subject must be followed by certain verb form, and so on, and so on. 

But as far as I remember, I learned very little on what I must say. What must I say if I want to congratulate someone for his achievement? or if I hurt my leg? or when I need someone's help? or when I am hungry?
Again, How to say it vs What to say (rules vs content), see the difference?

What could be the reason of this?
As an English teacher I have to admit that I like teaching grammar. Like here means REALLY LIKE IT. If I may choose, I prefer teaching grammar than writing, for example.

I believe many of the same profession as mine have the same feelings towards teaching grammar.
Why do teachers like teaching grammar?
Because grammar is easy to teach. They are rules. Black and white most of the time.
There are some grey areas or exception, but mostly black and white.
The rules say this, so do it this way.
And moreover, they are of manageable size and fixed most of the time.
For example, the rules of Simple Present Tense have been the same for the last century, I believe.

But teaching vocabulary.........?
There are so many words to teach. Where should we start? 
There are synonyms, there are idioms.....
and the number of words is increasing. More and more new words are 'invented'.
It is quite overwhelming for teachers, especially for us, local teachers.
How should we explain the difference between 'walk' and 'stroll', or between 'feel' and 'sense'?
Do you share the same opinion? Feel free to write a comment.

More is coming in Part 2.