I've got a similar set up with the soil very close to the surface. In the good old days of sodium chlorate weedkiller it was easy but now I get down and pull the weeds out. Boring but doesn't take long. I was thinking of trying a weed burner
It leaves a chemical film on the surface that prevents seeds germinating.
If you disturb the surface by pulling, raking or hoeing, you break up this film. You also bring new seeds to the surface and provide ideal conditions for new weeds to grow, which is why your previous regime suffered excessive regrowth.
Longshanks' neighbour does not keep disturbing the surface so it is hostile to seed growth.
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