True - the V2 documents were written more in the style of Papal Encyclicals, i.e., long and discursive.
So, no, I concede there was no implicit or explicit prohibiton on the State recognizing God and legislation based on Catholic morality and principle.
Yep. And in the Decree which actually bears more closely upon this question, Apostolicam Actuositatem, it says:
The apostolate in the social milieu, that is, the effort to infuse a Christian spirit into the mentality, customs, laws, and structures of the community in which one lives, is so much the duty and responsibility of the laity that it can never be performed properly by others.
So, how can the gov't make this "effective in practice"? Again, no solution or suggestion is found in this declaration
I disagree. It's talking about the other statements in the declaration about the freedom to be given to non-Catholics to worship publicly, and saying they apply even where special recognition is given to Catholicism or a false religion.
there's some vague suggestion in SC which makes the inaccurate implementation seem completely accurate
It is simply impossible to reasonably interpret SC's permission for the vernacular as allowing an all-vernacular Mass, for instance. If you want to talk about unreasonable or out-of-context interpretations, they exist for all other Councils, too (think of the Orthodox insistence that the Councils of Constantinople I and Ephesus anathematize the Procession of the Holy Spirit from the Son). DH is, when examined carefully, clear that its teaching doesn't effect prior teaching by Leo XIII, Pius XI, etc. on the Social Kingship of Christ.