I always thought ground cover hast to be weedier than weeds to get the job done. The job of ground cover is outcompeting weeds.
Great observation, but not necessarily. First up some definitions. A weed is a pretty loose term to any plant that is growing somewhere you don't want it. Different plants grow in different conditions, ie some germinate after fire (eg fire adapted shrubs like Hakea sp), some germinate in compacted soil (eg Dock), some in cultivated soil.
The ones we usually call weeds in gardens are the early colonizers filling the gaps in recently disturbed soil, or in orchards - they are the understory shrubs moving back in.
An orchard, where there is one tree crop, and little or no ground cover (in NZ, it is usually mown grass or sprayed out) is a very unstable system and requires alot of maintenance to keep it in that condition. It will constantly be invaded by plants to fill the ground cover layer, the shrub layer, the small tree layer and probably also the canopy tree layer because in many orchards the trees are planted so there are gaps between the trees to increase production.
Now, there is nothing "wrong" with an orchard, it just takes energy to keep it an orchard. Left alone, it will become a forest, and not necessarily have the plants we want in it.
To return to your statement "The job of ground cover is outcompeting weeds." I would say no, and it would be difficult to find one that did. A different way of excluding weeds, is to occupy all the niches that weeds could grow with plants that you wanted there - to plant a "forest" by design. This would mean there would be much less space for weeds to occupy. The only plants able to get in would be a special type of weed that I call an "Invasive weed": those plants able to infiltrate an already intact ecosystem - and the only plants I really consider weeds.
Geoff Lawton observed that in a Food Forest planted this way, you get fewer fruit off each tree than in a traditional orchard system. However the total yield measured against energy input is much higher, because - you are harvesting multiple species over the same area and the reduced weed interaction means there is much less maintenance.
I haven't experienced this myself - I'll let you know in 4-5 years.
By the way, check out that previous link on Nitrogen fixing plants. Do you have alder? That is nitrogen fixing, even though it is not a legume. Broom, gorse are both European nitrogen fixers. Eleagnus, not a legume but N2 fixer. I'm sure there are more - but I'm not familiar with Finland
(Please excuse the essay