Escalation die applies just to PCs.
Except when it doesn't. For instance, Dragons get to add the Escalation die as well.
But in general monsters start at a math advantage, and the Escalation die adds to the players to offset and then flip that the other way.
Correct. The Escalation Die exists as an elegant solution to the dominance of the "alpha strike." If you blow your biggest guns right away, you're at much greater risk of failing to connect, but if you do you'll take out more things early. If you wait, you have allowed the enemy to do more damage and survive more rounds, but you're noticeably more likely to succeed. Even after only three rounds (so the start of the fourth round), +3 to all rolls is a powerful thing, turning 50/50 odds into 65/35 odds (or, if you prefer, "every other attack fails" bumped up to "two out of three attacks succeeds.")
And the fact that dragons
also get the Escalation Die makes them
really scary, because now you DON'T have a force pushing toward victory, you have a force that pushes toward greater and greater danger for everyone involved. It situates dragons as a top-tier threat even if they aren't ancient wyrms, making their use and appearance appropriately weighty.
I like GM Intrusions from Numenera.
Oh God no, anything but that. "Mess with the players and punish them if they refuse" is in that rare breed of mechanics where it's actually
worse than critical fumble rules.
If you enjoy them, more power to you, but they're a horrible idea that actively invites DMs to be disruptive and divisive. That's the last thing we need in D&D.