It is not an inversion for emphasis. I think your confusion arises from the usage of the verb to unfurl which can be used both as a transitive verb and intransitive verb.
If it is used as an intransitive verb, the subject could be "the future" and the word order is not inappropriate at all. Your sentence can be rephrased to:
They were the quintessential American couple and the future was unfurling (or the future unfurled) before them like some endless magic carpet.
The sentence was shortened using "absolute construction" or "participle clause". The important thing to note is you can omit the subject of the second clause only when the subjects of the two clauses are the same as in:
Tom lost his keys (while) walking through the park.
= Tom lost his keys while he was walking through the park.
However, you can't omit the subject when they are different. In your example sentence, the subject of the first clause is "they" and that of the second clause is "the future". That's why "the future" is placed before "unfurling".