If we assert that aesthetic experience has no definable cognitive component what
Presumably it's only the philosophical interest which leads to the conclusion that it has no cognitive component in the first place ... Or rather, it's a matter of philosophical debate whether it does ... But if you are suggesting (as you seem to) that once a philosopher decides that aesthetic experience is non-cognitive there are no further philosophical issues, then I'll leave it to those specializing in aesthetics to provide an answer ... (At the least a non-specialist such as myself would wonder: if aesthetic experience is non-cognitive then how does it relate to other (sensory) experiences? what marks off an experience as aesthetic then? what is the nature/relationship of the different sensory modalities? the relationship between sensory experience and pleasure etc...?)
hope that's a useful start.