I tend to agree and you are right to state that the first two premises are uncertain....But, only if you are prepared to go beyond the observed universe...
It's not actually important to the argument but no, we don't actually need to do that. Quantum mechanics certainly predicts that some events have no specific cause. It provides a statistical framework but events such as one particular radioactive atom decaying at a particular time have no cause.
Relativity also makes it questionable whether the universe had a "beginning to its existence". Time, or rather space-time, is
part of the universe. If we take relativity seriously, the universe is a four-dimensional object that
contains time.
...At which point one does not have the luxury of claiming that the scientific method is good for all things.
Which matters not a jot. The point is that we have no reason to believe
any story that goes beyond the evidence, whether it involves a god, many gods, bands of magic fairies, universe creating spotty teenagers or none of the above, it's all just guessing.
As was stated before, most atheists are not arguing that there cannot possibly be a god, just that there is no reason to believe that there is.