I think that a rehoming fee is appropriate if you are going to the paper to find a home.
One thing is that it is like earnest money. Someone has to WANT the dog and have the money on hand to get him. It is more than that though. If you give me your dog for free, I answer your ad, and think I might like a neapolitin mastiff, even though I know nothing about them.
I get to your house and I see this gigantic, massive, monster looking thing with wrinkles and drool and droopy eyes. But the dog is nice and you seem pretty desparate for me to take him. So I get him out to the car.
Now I do not want this dog, not even a little bit. It is UGLY! But I just can't say no to you, because you are CRYING!!!
Well, what am I going to do. I take it home and do not like him any more the next day. I do not think I can face you again, and I do not want to drive all the way back over there. I will go to the shelter and tell them I found it alongside the side of the road. Or I will drop it alongside the road somewhere.
People who get something for FREE, well, they just do not think it is WORTH anything. They go to move, and they think nothing more of dropping the dog at the shelter. At this point someone is going to pipe up about their seven free dogs and how they would never... But we are talking in generalities. A major low-life scumbag that pays 1500 for a dog is going to try to at least get something out of him. If you pay $150, well, it really depends on how much $150 is for you.
I do not think you would take that neapolitin mastiff home because you aren't going to hand over the cash.
People drop dogs at shelters regardless to how much they paid for them. But I think that a dog that they were willing and able to pay something for has a better shot, maybe of going to a no-kill shelter, maybe of also being worth some training.
I think that sometimes people get a free dog and think of him as a free dog. What do you expect, the price was right, you get what you pay for, etc.