It’s also as often false as it is true. There is still such a massive demand for rescue dogs that people will pay a premium for them. There are foster households that are far from overfull. There are high-risk dog ranches that have room for more aggressive dogs, even human aggressive dogs.
Yes, there are times where you just cannot logicially find a match for a rescue. But there are also times where they put down a pet just miles away from someone who would foster or adopt them.
Unwanted animal euthanization rate in the US is arguably at an all-time low. And yet, PETA’s euthanization rate is still high, high enough that groups like the AKC are openly criticizing them for it.