Hi Ahmadmirza - some of those dandelions can be tough, and of course getting the doseage of the weedkiller right is tricky. I have a similar problem as when we moved here the gardens had been neglected for 20 years and now that I'm retired I have time to tackle it properly - this past year we've just been mowing to prevent the dandelions from seedng and infesting the neighbours' gardens.
I've got some weed and feed which I shall give a go in the spring. In the meantime I shall do as you and use a Daisy Grubber - they do work once you get the knack - but it could mean several days on my hands and knees on the lawn - OH might have to haul me back to the vertical after that
I know it makes a mess of the lawn, but if you do it in the winter, by the time you've fertilised it in the spring it'll soon perk up and the mess will be all gone. In fact, while the lawn's looking a mess it might be a good idea to give it a good rake and spike it all over with a fork to aerate it - then it'll be fantastic next year.
Alternatively you can use the 'paintbrsh and glyphosate method', but you've still got to pull them out when they've gone brown.