In defence of the Principle of Sufficient Reason I have always been entirely unimpressed by this kind of cosmological argument for God. For even if its premises are true - and they seem highly questionable - it does not imply God at all. Where atheism and theism are concerned, it does not tell more in favour of one than the other.
The principle of sufficient reason states that everything - everything, not just some things and not others - has an explanation for its existence.
That principle is by no means obviously true. For after all, if it true, then it generates an infinite regress. A cannot be explained by A, and so B has to be posited. But B cannot be explained by B, and so C has to be posited. And on and on for an actual infinity.
So, it seems false upon reflection. And if one thinks it is not false, for one thinks there is nothing problematic about there being actual infinities, then it does not lead to God, but to an actual infinity of prior explanations. God isn't in the picture at all.
If one modifies the principle so that it is only contingent existences that require explanation, then all you get to conclude is that there exists at least one necessary existent. But there's no reason to suppose that necessary existent is God. That's like concluding that becuase 'someone' clearly shot Kennedy, then it must have been Mrs Smith at 28 Acacia Avenue.
In fact, it is worse than this. For it seems self-evident that we - minds - are not necessary existences, but contingent ones. I am a contingent existence. So why assume that a necessary existent would be a mind? Minds seem no more or less contingent than physical stuff, and so to suppose God - a person, a mind - is the necessary existent is not just a huge leap, but an implausible one.
And it gets worse still. For consider this. Necessary existences confer necessity on what they cause. And thus a necessary existent can only explain other necessary existences (and it woldn't really be an explanation either, as by hyothesis if something is necessary, it does not need explaining). To see this note that a necessary existent either contingently causes something else to exist - but if it does that, then it was contingent whether it would cause it or not, and thus no explanation is provided. In order not to generate a need for explanation, the necessary existence must be supposed necessarily to cause what it causes. But if it necessarily causes what it causes, then what it causes to exist, exists of necessity as well. Yet necessary existences are supposed to explain contingent existences! They can't - as nothing a necessary existence explains will be contingent, but will be necessary too.
Necessary existences are actually quite useless explanatorily, for all they do is confer necessity on what they cause, but as what needs explaining are 'contingent' existences, they are useless.
But again, even if they weren't useless explanatorily, there is just no reason to suppose any necessary existent is God and positive reason to think it wouldn't be, given that all the other persons of our acquaintance are clearly contingent existences not necessary ones.