You're correct that Archimedes presaged core concepts of calculus: in his work building upon the method of exhaustion, quadrature for certain conic sections and tangent of a spiral. It's nonetheless quite a ways up the tech-tree from calculus. Important developments were also contributed by al-Tūsī in the 12th century and Kerala school mathematicians (Indian, prominently around 14th century, though their broader influence is uncertain).

As advanced as Ancient Greek mathematics was, its heavy emphasis on geometric formulations made many developments more difficult. The Hindu-Arabic numeral system, algebra, invention of a zero number concept (Cardan's proof of the cubic and quartic would have gone significantly easier if he had utilized zero as a number) and Descartes's Analytic geometry were important developments on the way to calculus. By the time of Barrow's and Fermat's contributions, nearly all essential components of calculus existed, just requiring some genius to grab, synthesize and streamline them.

It's interesting that calculus was independently invented 3 times nearly simultaneously (AFAIK there's a lesser known one). Yet, I don't know any reason for it not to be invented 3 centuries earlier.