I have read the two articles in question by Vilenkin (and coauthors) and it might be relevant to state what the articles show and do not show because this is often being confused as it is on this thread.
What Borde, Guth and Vilenkin showed (and the proof is not in dispute) is that if you start out with a classical space-time, and you assume it has always been expanding, then at some point in the past at least parts of the space-time will undergo a singularity, i.e. our description will break down.
To put this in lay-mans terms: Suppose we ignore quantum mechanics and assume the universe has always been expanding, then at some point in the past our current laws of nature won't work anymore.
However I can't think of any cosmologist who would think that we should not take quantum mechanics into account when the universe is the size of an atom, and the BGV theorem does not do that because we don't know what that description should be. This is not me saying they are mistaken or ignoring something, they say so themselves in the paper!.
To use the BGV theorem to argue for God is no better than to say: "Oh gee, a simple model of the universe we know is wrong can also be proven to break down, therefore god must have made everything".
And it's even worse than that. In reality, some scientists believe the universe had a "beginning" (but this certainly include natural beginnings), and some that it did not. Even if we trust the BGV theorem despite being based on a known incomplete model of the universe that would still only leave us with those models of the universe where there is no beginning. it does not get us anywhere.