Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.
So, which is it? To apportion the blame, I suggest that the fact of observation be that “errors” are independent and identically distributed with finite variances. Observers may as well vote for Lyapunov’s set of conditions. I am sure that mathematicians are more than happy to shoulder the responsibility for the resulting Central Limit Theorem.