musser zoger: I believe it’s the Rambam who says there will be two parts to the times of Mashiach. In the first part, life will continue as it does today, with one big exception – it will be clear that Hashem is the only God, and everyone will accept that. We’ll have a Beis HaMikdash, and will do avodah, but there will still be a Yetzer HaRah, and we will (likely) still do aveiros. It will be comparable to the times of the 1st Beis HaMikdash (except with no other religions).
After this time comes Acharis HaYamim, which is when it says Hashem will slaughter the Yetzer HaRah and the Malach HaMaves. It seems that the time referenced in the davening is the 2nd time the Rambam mentions.