Logic encompasses science, not the other way around. I’m sorry to have to tell you that your logic is flawed.
We both accept the Big Bang as fact.
The truth of the conservation law is not self-evident—rather, it is validated only by empirical methods.
To say that it is not self-evident is to say that it cannot validate itself. The law tells us nothing about whether it has always existed or whether it appeared at some moment for the first time, so your assertions are meaningless.
Bless your heart. I hope you're not too distressed over it.
"The law tells us nothing about whether it has always existed or whether it appeared at some moment for the first time
That's right if you're referring to when it was first realized. Of course nature doesn't care whether anyone realizes how it works. Nature doesn't care what anyone thinks about how it should be, it just is.
When you have evidence that energy can arise out of nothing and the law is invalid, you can present it. Until then the law stands as it always had, whether folks realized it, or not.