I can’t believe people actually hand weed their lawns. You’ll never get all the small things out. And they’ll just bounce back.
The KEY thing is not to overdose. If you over spray an area, even once, the grass is likely to die off. Broad leaf herbicides depend on surface area of the leaf. Grass ha a a thin leaf so absorbs less. One spray won’t affect it. Two will.
I have been using Relay turf the past few years. A single spray kills all the weeds for the season pretty much. Granular complete products depend on a granule happening to land on a leaf and staying there until it gets wet and dissolved, so obviously they miss a lot of the smaller ones. And the bags of evergreen are only 13kg from memory?
If you don’t want to buy a proper sprayer, just get someone in (NOT greenthumb!). I’d typically charge £30-40 for a 100m2 lawn, weed free for a year. Very few weeds year 2.
Personally, I’m going the other way and trying to create a rich and interesting garden for my kids, which will have clover, dandelions, moss and daisies!