There is nothing wrong with a mongrel as such. I have one. He's ace. The problem people have with them is the way they are bred.
There are far too many dogs being bred and not nearly enough suitable homes for them all. Responsible breeders breed only to improve the breed. They often wait until they have a waiting list or only breed when they want another dog to train/show. They'll only sell their dogs to responsible homes and will do their utmost to ensure their pups never end up in the pound, including taking them back in themselves, even ten years down the line. Unfortunately these breeders are few and far between.
The problem with mongrels and designer breeds being bred is that a) there are far too many being bred. b) the breeder is in it purely for the money and will sell their pups to anyone who is willing to pay, no matter how unsuitable that home is. c) the dogs are not health tested prior to breeding resulting in sick puppies.
The 'myth' that mongrels are healthier, is just that, imo. A myth. When you breed two different breed dogs, unless the parents are health tested prior to breeding you take a massive risk that the pups will inherit the genetic faults from both breeds. Breeders of mogrels and designer breeds very rarely health test.
There are two in my family. My own ex poundie, who has a very expensive skin condition that probably could have been prevented by correct breeding and my sisters bull lurcher (a bull breed/greyhound x) who has a hernia that's going to cost them a fortune at some point and is not insurable. That could also have possibly been prevented by correct breeding. The 'breeder' couldn't sell him because of the hernia and so was going to put him in the free ads as 'free to good home' (read free to the first person willing to take him off my hands, no matter how unsuitable they are) my sister heard about him via a friend and stepped in and took the poor pup home. He now has his own bank account to save up for the op he is more than likely going to need in the future. He's lucky. A lot of other people would pts before shelling out hundreds of pounds on an operation. A lot of other 'breeders' would have just drowned him at birth.
When you add in the designer breeds and the popularity of them vs the lies told about them i.e non shedding, easy dogs etc. rescue have a massive problem on their hands finding space for these dogs, all the while unscrupulous BYBs and puppy farmers are pockting money like it's going out of fashion while their poor bitch is bred from again and again and again, normally untill she dies or becomes to sick to breed.
You cannot get a responsible mongrel breeder. It's impossible. There is no responsible reason to bring yet more dogs into this world unless it is to improve the breed, even that is a questionable reason, given the amount of dogs we kill for lack of good homes each year.