(My comment below this one was meant to be directed as to the existence of speech laws, not the US constitution's position on 'natural laws')

It's basically wrong to suggest that somehow the US's constitutional assertion indicating certain rights are 'natural' is somehow problematic - or even unique.

The 'existential and inalienable rights of human beings' have been recognized not only in constitutional and legal forms throughout the world, but in various other texts and declarations.

As I mentioned, the Canadian constitution refers directly to divine providence. (Which is another way of saying 'natural rights')

The 'Basic Law' in Germany starts with this text:

"Conscious of their responsibility before God and man,
Inspired by the determination to promote world peace as an equal partner in a united Europe, the German people, in the exercise of their constituent power, have adopted this Basic Law."

This is definitely an existential assertion.

The UN Universal Declaration of Human Rights starts like this:

"Whereas recognition of the inherent dignity and of the equal and inalienable rights of all members of the human family is the foundation of freedom, justice and peace in the world"

So they refer to inherent dignity which is essentially another way of saying 'natural right'.

The Constitution of the Russian Federation:

"We, the multinational people of the Russian Federation, united by a common fate on our land, establishing human rights and freedoms, civic peace and accord, preserving the historically established state unity, proceeding from the universally recognized principles of equality and self-determination of peoples"

Once again universally recognized principles yada yada ...

Poland:

"We, the Polish Nation - all citizens of the Republic, Both those who believe in God as the source of truth, justice, good and beauty, As well as those not sharing such faith but respecting those universal values as arising from other sources, Equal in rights and obligations towards the common good - Poland,"

Greek constitution starts like this:

"THE CONSTITUTION OF GREECE In the name of the Holy and Consubstantial and Indivisible Trinity"

(Admittedly, it doesn't quite specify those things as sources of rights or provenance)

Not surprisingly, most Arab state Constitutions defer to 'God' as the source of their ultimate authority, but again maybe unsurprisingly, don't dwell a lot on 'rights'. But from the 2012 Syrian constitution: "Freedom shall be a sacred right" - again alluding to the transcendental nature of certain rights.

Shall I go on?

Note that Australia (the nation in question here), and a whole host of other nations which do not specifically hint at the existential nature of human rights in their own texts, are in fact signatories to the UN Universal Declaration of Human Rights - which legally enshrined the quote above. So that's going to add Sweden, UK, Norway etc. etc. as well to a very formal declaration of the 'natural rights' of people.

So finally - the US's assertion of 'natural rights' is a) not an odd, special or contentious thing b) common enough in the world, constitutionally and legally that we can't really say that this nature of their Constitution is very unique.