It’s a bit difficult to determine the best place to start with this type of discussion. Are the constants we see today the result of random interactions that occurred prior to the emergence of our universe? That would have to be settled at some point.
As to laws of nature, those laws require random interactions in many cases, such as thermodynamics. Decay constants for isotopes describe random decay. Laws of optics dealing with diffusion of light require random absorption and emission of photons.
As heddle mentions, there is also the hindsight nature of fine tuning. You can get very different results from very small changes in the history of a process, but that is true of many random processes. Just a small nudge here or there in a ping pong mixing machine and you get a very different lottery outcome. From that standpoint, each and every lottery result is fine tuned, but is still the outcome of random processes.
There are a lot of entry points we can take, and I am not sure which one you think is most important.
In general, my experience in these discussions is that people tend to get probabilities really wrong (present company excluded, of course). One common mistake is the Sharpshooter fallacy where probabilities are calculated after an event has already occurred.
How does science point to its creator?
@heddle wrote a great post, so I would start with that one. We first have to define what exactly we mean by fine tuning. If fine tuning means that very small changes to constants and starting conditions would have produced a very different universe, one that probably isn’t capable of producing life, then I would fully agree. It really isn’t an argument, but rather a statement of fact. I see nothing in that statement of fact that I would criticize.
It is the jump from fine tuning to the claimed obvious need for a fine tuner that I disagree with. While a fine tuner could exist, you need more than just a possibility. You need some evidence (of the type I have described in previous posts) if I am going to be convinced that there is a fine tuner.