So, an interesting consideration: the Promethium Mine is a Support, which the Infantry Conscripts get +2 per Support. The Mines do not self-trash themselves after the resources run out on them and are not unique, so I'd think a 3-of that card wouldn't be bad for them. Plus more resources never hurt.
Militarum and Mines
You are correct about the way the Mines work and that they continue to power up the Conscripts after being depleted. Whether of not it's worth putting 3x in the deck, with or without the Conscripts, might be more a question of play style.
The real issue with Promethium Mine is that it takes so long to gain a true resource benefit from it, 3 turns even. In a game like Conquest you just don't have that many deploy phases to earn back your investment. But I do agree that they work well with the Conscripts so perhaps the PM is more about the Astra Militarium deck after all? They at least have an alternate use for the mine. I'm so disappointed the mine doesn't have even 1 shield on it so it would not be utterly useless if you draw it late in the game, LOL!
True about late game mines. I think something that will need to grow and explore is the mulligan hand. My other LCG is Netrunner and it's finding that balance between econ and ICE or programs. If you're going 3-of PM in AM, you may want to mulligan to get one or maybe some other supports in hand early. In other decks, you want cheap Army units to spread out. It'll be a growing part of the game as we get more cards.