No way, no how could you give a dog away for free here and I cringe whenever I see people trying to do so. They end up with dog fighters as bait. Cats will too.
We watch our dogs like hawks, I'm terrified of someone stealing our Pit. There have been so many in our area that have gone missing. There was even one case where someone broke into a womans house. The only thing missing when she got home were her two Pits, their leashes, and collars. Her lab was locked in the bathroom.
I don't see a rehoming fee as charging someone for doing you a favor. It simply means that you are guaranteeing that the dog gets an actual home. Especially for a Corgi (not going back to look but I think that's what this dog is?). A free purebred corgi would be a goldmine for a puppy mill. Any popular purebred dog would.
On a different note, there used to be a guy in town that was always posting classified ads with baby ball pythons for sale. in the ads he would usually add that he ran a small animal rescue and would gladly rehome your hamsters, etc.