The north side of the house has a strip of earth that struggles to be kept in lawn. In fact, if you remove the noxious weeds, there would not be much. So, I would like to explore groundcovers. The area can get very dry. House is 3 stories so mainly shady though it can get some sun in summer when sun angle is higher. I would like for the groundcover to be able to withstand some foot traffic and not get high enough to have to mow. And since I want EVERYTHING...flowering would be nice so pollinators would have another nectar & pollen source. Evergreen?

Does such a beast exist?

You could do a mix of Sedum album, dasyphyllum, anglicum, hispanicum, and lydium. These are all small ground cover type sedum. I would think by using them all you would find which fit the site the best, as the hardier ones would flourish and the others die out. Having a mix might also add to the visual interest, especially if you place path stones.
All of these bloom and are evergreen.
There may be others that would work also. Lets see what others come up with UrbanWild.
I love your idea.

Since it is shady don't know how well the sedum would do unless it is bright shade.
Vinca is one I use, you have to water for first season until established but then it'll take off. There is a post on NGA about growing in shade but can't remember where, have to look around. I like the false strawberry but they can get out of control. They get a yellow flower then a small red berry that isn't poisonous just small and not flavorful. Can't remember what it's called
For dry shade areas looking at "weeds" you like, and planting more of those works too.