The song aspect struck me more as metaphorical, like the siren calling out to Ulysses in the Odyssey. Any actual "Machine" is really just an obsession dragging the character onward towards their demise, but not that of The Machine. Just my interpretation.
For the more practical sake of playing, I saw it as if one of us (as a character) were to find Da Vinci's notebook containing sketches for a Time Machine to bring the great artist into the future. A brilliant invention, but one that never saw fruition. The first person to find this outline tries to pick up on Da Vinci's work but fails. Someone else finds the notes of Da Vinci's successor, and proceeds on with their work, but also fails. And so the process continues until the game ends when there are no more players.
This is how I understood it.
I ran a game like this for myself called The Sealed Library. (Solo-gaming and PbP have been good to me during the pandemic.) You keep a log until the game ends, and the character meets their demise. It was played with a deck of cards and a Jenga tower. When the tower falls, the game is over. You basically wrote a journal entry prompted by a card drawn from a deck, then pulled pieces from the tower. You pull the number of pieces from the tower equivalent to what you rolled on a d6. Roll a 1, take one piece. Rolle a 6, take 6. You get the idea.
It's not gamey in the traditional D&D sense of the term, but as characters, we take part in a simulation of sorts. It is very much a shared writing experience, though, that's definitely my understanding if The Machine as well.
While I was playing this I thought it would translate to PbP really well because it's just writing. Since you can play the Machine by mail, I don't think it would be an issue in a PbP forum. I'll put up two other threads to put some bits of the other games up so you can get an idea of what they look like.