No ideas sorry! I have the same problem, this time of year the weeds are growing at an alarming rate - way faster than I can pull them out. The old adage goes 'one year of seeds = seven years of weeds' or something like that. Removing the weeds before they flower and set seed will save a lot of angst in the years to follow, even if you just cut the tops off them. Many of my gardens are near a neighbour who cares not a hoot that his back paddock is harbouring weed infestations that could take over the world. Add to this the weeds that come from the surrounding pasture, grass, clover etc.
I would love to potter through my garden with a tasteful trug and immaculate gloves, pulling a juvenile weed here and there out of beautifully friable, compost-enriched tilth. My reality is never this!
I have to take a pretty hard stance. Although I don't use sprays of any sort on my lawns, roses, or veges (a copper spray on the fruit trees very rarely) I do use glyphosate. Everywhere. I have been known to use it IN my garden beds when faced with a particularly beligerent weed. I spray the boundary, squirting out into the neighbouring properties if I see weeds yonder. I spray the edges of the beds to keep the mad-hatter kikuyu grass from smothering everything. I spray the weeds that colonise the gravel driveway - all 500 meters of it. I spray around the base of the trees that grow in the lawn to make mowing that much easier and to stop me ring barking the trees with the edge of the mower. I spray in the tree boxes (that protect the trees from stock) because the horses can't reach in to eat them and they go to seed quickly. I spray the paddocks to kill weeds that reduce the grazing quality.
With all this spraying you'd think that the property looked like it was dying of some pox, with browned-off bits everywhere, but actually it never looks too bad probably because I don't do it all at once.
The other thing I do is mulch. Big time. I have a lot of semi-broken down stable muck and a massive pile of wood chips. You are right, this is too coarse for tiny seedlings. And of course it stops any nice plants from seeding about too. I tend to take cuttings or dig up tiny seedlings to grow on a bit before planting them back in the garden. I apply a massively thick layer. It is great for water retention in summer, really makes a difference to the weed population, and eventually becomes incorporated in the soil. I have had to resort to weed matting in places. I am about to re-do a path that has every weed known to man in it, the soil sems to have a never-ending supply of seeds too. I will take up the massive schist pavers, pull out the clumps of black mondo, de-weed it. Then re-plant the mondo in clumps, lay weed mats around the plants, put gravel down and replace the pavers. Might even do it today...