First of all, let us define the so-called supernatural machinery. Here is a helpful quote from a letter Alexander Pope added to the second edition of the poem:
The Machinery, Madam, is a term invented by the Critics, to signify that part which the Deities, Angels, or Dæmons are made to act in a poem: For the ancient poets are in one respect like many modern ladies: let an action be never so trivial in itself, they always make it appear of the utmost importance. These Machines I determined to raise on a very new and odd foundation, the Rosicrucian doctrine of Spirits.
Thus Pope is mocking both ancient Greek and Roman poetry, much lauded by the Romantics and the major source of inspiration for European poetry (the poem is an epic, an ancient form if ever there were one), and the pettiness of his contemporaries. He is ridiculing the concerns of his fellow Englishmen by making a tiny incident take cosmic proportions, all through his sylphs, a satirical take on the traditional machinery.
It is also helpful to take into account Pope's Catholic faith and its ostracized status in Britain at the time to understand what he is getting at. Mocking tradition and inventing substitutes for the gods is also a way to rebel against the hegemonic Anglican faith that forbade Catholics from taking part in much of public and political life.