Here is my view on this.

I would agree that the human brain does not appear to have hard wired grammar rules for any particular language.

There has been some speculation, with some evidence to support it, that it may contain some sort of generalised grammar rules with parameters of some sort that are then set for the particular language being spoken, but I'll ignore this here.

To learn a language would probably require a very capable modelling system. The brain is clearly very good at this sort of thing. Furthermore, the brain does not just need to look at words on a page. It can relate words to what the eyes see people doing for example. As a very crude example. If it experiences pain commonly after someone says "I will hit you" it can start to get a grasp of what a future tense is.

I think that grammar rules would have to be programmed into computers for now, as a substitute for any decent sort of learning ability in computers. As computers get more sophisticated and we understand more about how to make machines learn they will just pick up language as we do.

Could neural networks learn grammar? Possibly. There have been some experiments. e.g.

http://www.amlap.org/2001/proc/proceedings_online-node23.html

The English in your post is very clear, by the way.