The distinction between “apparent age” and “embedded age” does not solve the scientific problem.
Science does not decide whether miracles can occur. Science tests physical explanations for physical evidence. Once the claim is about rocks, isotopes, fossils, stratigraphy, cooling histories, or the age of the Earth, it has entered the realm of observable physical evidence.
If “embedded age” means that rocks were created with isotope ratios, daughter products, crystal histories, geological layering, fossils, and other physical indicators that normally require long periods of time, then scientifically that is still an apparent history unless those events actually occurred.
Calling it “embedded” does not change the issue. It means the object contains physical evidence of a history. The question is whether that history really happened.
A rock is scientifically old if time actually elapsed since the relevant event, such as crystallisation, cooling, deposition, burial, metamorphism, or erosion. If it was created recently with physical properties that indicate billions of years of elapsed processes, then it is not scientifically old; it has been made to look physically old.
So the issue is not that science fails to distinguish miracles from natural processes. The issue is that a miracle claim cannot be used as a scientific rebuttal to geology or radiometric dating. It can only remove the claim from scientific testing.
In that case, the argument becomes theological, not scientific.