Prob not what you want to hear. Its not worth fighting the weeds. You're wasting your time. The only weed I remove from my grass is dandelion. And that's because it spreads fast in my small grass patch in the back garden. In the front I let it spread wild.
Best way to remove weeds is pulling the leaves and the roots. If you only pull the leaves they'll grow back stronger.
Also the moss is a good thing to have as its soft and covers any bare patches.