Stylistic usage of "of"

Solution 1:

The "of" is a simple qualifier denoting what the comparison is between.

I have the highest salary of people in my family.

means that comparing all the people in my family I have the highest salary.

The Americans have the fullest poetical nature of all the nations.

has a similar meaning.

The sentence you are reading simply reverses the order:

Of all the nations, the Americans have the fullest poetical nature.

I prefer the version with the comma, but the version without means the same. The full sentence just claifies which nations we are talking about (all at any time upon the earth) and adds the qualifier "probably".