Dandelions are really, really hard to eradicate by pulling. The roots grow very deep, and if you don't get them completely, the plant can re-grow from what's left.
Even if you do successfully get it out, it really is going to be more work than painting a weed killer on them.
My dad use to have my brother and I work for hours during the summer pulling dandelions in the lawn (to be fair he was out there with us doing it himself also). We each had a knife with about a 4" long blade, we would cut the root as deep as we could and pull the top out. Never really seemed to reduce the number we had.
Even if you do successfully get it out, it really is going to be more work than painting a weed killer on them.