Cutting will control some weeds, encourage others. Thick healthy grass, well fertilised and growing strongly, will discourage the weeds that thrive in poor soils but will simply make others grow faster.

If you damaged the lawn with your weedkiller then you used it wrong. Sorry but it had to be said. Lawn weedkiller, used according to the directions, doesn't kill grass. In my experience it doesn't even make it temporarily yellow although that can happen. I find the granular weedkillers to be very unreliable. I use a soluble weedkiller, which one doesn't make a huge difference except for a few resistant weeds. I apply with a spot sprayer since I now have only a few weeds. I make up my own mix from a concentrate since going over the whole lawn may take an entire spray bottle. Try to spray on a still day so you don't spray all your borders, a nice cool dry day when the soil is moist so that the weeds are growing strongly. Make sure it doesn't rain and you don't water since that will just wash off the weedkiller.

Digging can help. Weeds like dandelions must have the taproot completely removed or they just grow back. The taproot will be mant inches long on an established plant. Others have a spreading habit and any pieces that you break are likely to just start new plants. It can be fun tracking each piece of weed to the stem where it reaches the ground but its not everyone's idea of entertainment. Many weeds thrive because they produce masses of seeds, which will continue to germinate for a number of years. Many are very mobile and will blow into exposed lawns from neighbours gardens or wild areas. Always try to remove weeds before they can set seed. Spraying a weed may not stop it from setting seed so try to at least cut off any flowers.