By applying the logic that phenol exists in enol form rather than keto form as it attains aromatic character, why is it not the case in coumarin?
The answer is as simple as it will be unhelpful to your original question: There is no tautomer possible.
Coumarin does not have a keto group, and there are no enolisable protons either. Coumarin consists of a benzene ring, connected to a heterocycle with the functional group lactone. The molecule is flat and the conjugation extends throughout it.
What you have drawn are two resonance structures. Both of them are valid configurations (there are more), but neither actually exist; see What is resonance, and are resonance structures real? In both of your drawn structures the nuclei positions (in the clamped nuclei approximation) do not move.
The concept which applies here is completely different from keto-enol-tautomrie, where there are two different, observable molecules.
The question which you are probably asking is: Which of the drawn structures is the major contributor to the (so-called) resonance hybrid.
I have run a quick calculation of the DF-B97D3(BJ)/def2-SVP//GFN2-xTB level of theory, which suggests that the conjugation is skewed towards the lactone group. More investigations would be necessary, but the π orbitals should give you a general idea.