The Kalam Cosmological Argument

The Kalam cosmological argument points to the creation of the universe as proof of the divine. To do so it involves proving that the universe did indeed have a beginning, and has not existed for an infinite amount of time. It also argues that all things including the universe itself, with the exception of God, are caused. Furthermore this cause was personal.
The first argument presented is that an actual infinite is a logical impossibility. Allowance is made for potential infinites (i.e. the future), but it is differentiated from an actual infinite, such as the case would be if this universe had traversed from an infinite time to present day. The analogy drawn to prove the impossibility of an actual infinite is that of Hilbert’s Hotel. A mathematical theory that demonstrates the trouble with relating infinity to existence. Adding, subtracting, indeed any arithmetical process does not apply to the infinite. Infinite plus one is infinite, minus one-hundred thousand is infinite, so on and so on. So how can anyone/thing, including time be counting down from infinite? At what point do you start counting in real numbers? How is it that the counting hasn’t stopped yet at any (or every) point in history?
If an infinitely old universe is impossible then it must be finite, that is it must have had a beginning. Now, why did this beginning come to be? This theory then presupposes that, as per our experience, the universe must have been caused by something. The only something with the ability to cause without itself having been caused is God. There is no cause outside of time except god himself. Therefore only God could have created the universe.
I think this theory makes perfect sense in the way we perceive causation, but doesn’t it also seem likely that our modes of perception are not up to the task of discerning such things as the ‘birth’ of the universe? How likely is it that birth is a term that doesn’t apply here at all, if only we could truly understand it? There is