If it didn't have a beginning, then as the article says, it would be eternal and would have have burnt itself out long ago.
This is the trouble with relying dishonest creationist propaganda sites. You are treating the problem with a rather quaint Newtonian view of time and then adding the more modern second law of thermodynamics to it until you get an answer you like, while ignoring all the other applicable modern science and, in fact, basic logic.
I've already given an example of how the universe might have a beginning (in the sense that we can only extend the time-like past direction back a finite distance) without requiring a cause (
#921). There are others and there are others still that suggest something might indeed be 'eternal'.
The point is not whether any of these are right, the point is that (according to what we know)
any of them
might be. Hence your 'logic' is undone by counterexamples.
More generally, even if we accepted your assumptions (momentarily), we instantly see that they must be wrong in some respect, because the universe exists and hasn't "burnt itself out". Logically, that is all we can deduce, there is no justification for the god fairy tale.