You are mixing, as I am somewhat as well, scientific assertions with logic. We should not go willy nilly back and forth -- especially into logical arguments -- without turning on the reason light, and saying so. This following is a physical argument.
You state that time is necessary for beginnings and endings. That is not in evidence in our argument. If time is a measure of beginning and endings, it is not necessary, it is just observed or, past it, intuited. This does entail that there could be a pre-time existence of something. In the ultimate black hole fore example, containing all the matter of the universe, time could possibly be stopped, as gravity could be too great for time to pass. Remember, Einstein showed that time is tied to gravity, speed and energy. It is non separable. Time is just a measure of change. Before change, existence can be, but is timeless.
The conundrum of time before time is a problem that you made up. I can logically say, before time began, and existence was not as we know it. From the inconceivable we can inject a temporal mode of argument into a "time" before time. Of course we cannot step too far before time in reality, as it does not exist. Perhaps someone tried to do that, and the subsequent illogical time paradox caused the matter thing to explode :]
Einstein also showed that time can run backwards for some particles under certain conditions.
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