Is there any rationale to prefer, in academic fields of art, Continental philosophy over Analytic philosophy? I guess there might be a historical reason why philosophy evolved into two approaches but why I do not hear in my art studies anything about analytic philosophy of art as if it never existed, and only hear about Barthes, Foucault, and their like?
There's probably no way to summarize this without drawing hate from both sides, but here you go:
Analytic philosophy tends to approach meaning as something to be discovered, whereas Continental philosophy approaches it as something to be created (or co-created) by human beings and/or human society. From the Continental perspective, therefore, artistic expression is vital. From the analytic perspective, art is one of the things to look at and/or dissect.
Because the Continental Tradition continued to be concerned with what art is and cultural issues, and the Analytic tradition largely switched to modelling philosophy after science and focusing on logic, sidelining culture and aesthetics. This is linked to a difference in perspective about what philosophy is for, and what constitutes an authoritative public intellectual, I'd say at root.
Thinkers who actually read widely in both traditions, like Rorty, often reject it's continuing relevance. I would describe the distinction as a tool for talking about the history of philosophy more than anything.
Key thinkers who are considered 'Continentals' come at their work through the Critical Theory approach, which:
"focuses on reflective assessment and critique of society and culture in order to reveal and challenge power structures. With roots in sociology and literary criticism, it argues that social problems stem more from social structures and cultural assumptions than from individuals." from Wikipedia
Embracing the postmodern condition, and accepting the end of universal objective truth, is pretty much limited to the Continental Tradition. And
I would relate postmodernism to art criticism as the shift from what art IS (implicitly a metanarrative, making a claim to be the dominant interpretation in cultural discourse), to what art DOES, for it's different audiences, for it's maker, considered in different contexts, in layers of abstraction and reference.
Naturally postmodern thinkers also apply this to what philosophy is, and is for.
See SEP's article on Rationalism vs. Empiricism. This is a centuries old dispute over whether philosophy should focus on evidence of the senses or on introspection and rational analysis. Empiricism was primarily carried by the anglophone world (the UK and its colonies and offshoots); rationalism held sway in European universities.
Empiricism isn't well-suited to the philosophy of art, since art is (by its nature) introspective and expressive. Most artists are concerned with issues like aesthetics, cultural significance, the nature of metaphor, allegory, and symbolism, and other matters that aren't directly attributable to sensory evidence. While I'm sure that empericical analysis could be used within a comprehensive analysis of art, it doesn't surprise me that philosophers of art and the professors who use them give it short shrift.