I think there are two issues, for a terminally ill animal that is loosing a battle to survive and a animal that has suffered trauma. A desperately sick animal I'd go to a vet and get it done with drugs and Anastasia. A trauma case is really the hardest one here, a animal that has suffered a horrible injury and is suffering (crushing injury of core for example) taking this case to a vet is not really humane it simply needs a swift death. In a case of something severe the swiftest death that you can manage is better than a slow agonizing one. If you simply cannot pith or smash the head freezing is still better than days of agony. I have been forced to end the suffering of an animal I cared about in that case it was a cat that had been run over and still alive with the back end crushed flat. It was suffer or kill her. I used a pocket knife to pith. I cried and vomited and scalded my hands trying to wash them in way too hot water. If I had been asked before if I could do such a thing I would have said no way but when push came to shove action was required and death could not end the pain fast enough I used what was at hand and did the best I could. Was it one of the most horrible things I have seen and done YES. Would I do it again... I think so. My point is in the case of agony going and finding things just isn't helping use what you have and get it done now not later. If the best you can do is the freezer that is the right answer. 20 mins is better than 20 hours.
I hope that you are never faced with this decision the last time this topic came up I had less experience. I hope the next time that it comes up everyone reading this one will not have any practical experience. Knowing that you did the humane thing does not make me feel any better.