I would say that such adverbial phrases (that is what I would call them) indeed function exactly the same way as adverbs. Whether they can modify an entire sentence or not is a matter of semantics (of the adverbial phrase) rather than the form (having multiple words instead of just one). Consider:
Rather surprisingly, the sun had already set.
It is clear that "rather surprisingly" has exactly the same grammatical function as "surprisingly". So one has to look at the semantics rather than the form. Here, "rather" modifies an adverb and yields an adverbial phrase with the same grammatical function. Since "surprisingly" can modify a sentence, so can "rather surprisingly". Same with:
Not surprisingly, the sun had already set.
Also, in English and many other languages, prepositional phrases can function both as adjectival phrases and as adverbial phrases, and yes, they can modify entire sentences just as adverbs can. Also, they may not always occur at the front or back:
Grammar is in fact very complicated.
The "in fact" here modifies the entire sentence, but euphony allows us to put such short adverbial phrases just after the equative verb "is".
I would not consider these adverbial phrases to be always disjuncts. Consider:
In the mind of a good ruler, nothing is more important than the people.
I think it is clear that this adverbial phrase modifies the entire sentence, making it an assertion about what is in the mind of a good ruler. Without the adverbial phrase, it would be an assertion by the author of the sentence, which is a different assertion altogether. One can imagine a king writing:
In the mind of a good ruler, nothing is more important than the people.
But I regret that I cannot be a good ruler. I cannot sacrifice my child for the people.