Clause? As I parse and label it, there isn't one. Still, we should be able to create a couple:
Now that that has been said, I can also say that YouTube has become a very different place in recent years.
The structure "that has been said" is a clause. The demonstrative "that" is our subject, and (in the original context) it indicates the content of the preceding paragraph. The predicate "has been said" employs the present tense, perfect aspect, passive voice and indicative mode. On its own, the clause works as an independent clause and could stand as a complete, separate sentence.
We don't want an independent clause here.
The phrasing "now that that has been said" is a subordinate clause, specifically an adverbial clause. In my example, it acts as a supplement to the matrix clause "I can also say that...."
But, why should we use a dozen words when we can simply use two or three? Why should we deal with a layer of indirection (as represented by the orienting matrix "I can also say that...") when a direct statement is simpler and easier to understand?
The absolute construction in the original example accomplishes all of that.
In the original example, we still have the demonstrative pronoun "that" indicating the content of the preceding paragraph. We have the participial phrase "being said" as a post-positive modifier of this pronoun. The resulting three-word phrase is a supplement of the direct independent clause of the original sentence.
In my framework, "being said" doesn't count as a predicate. It has no tense. It doesn't attach to a subject. It's not part of a clause. In contrast, the Wikipedia entry for absolute constructions does use the label "non-finite subordinate clause". Other sources still use the traditional label of "absolute phrase". Either way, the principle and usage remain the same.
There are options. "That said", "that being said", and "that having been said" all work as valid and sensible absolute phrases. Choosing between these is a matter of style, emphasis and dialectic preference. None of those options are more correct than the others, but different native speakers might find any one of those options to be the more natural choice.
By the way and despite what Wictionary says, none of those options have any tense, so none of those options can imply simultaneity of the action. They simply imply relevance of the resultant state.