Where did the writers, director, the film say that the Kelvin was an experimental testbed unlike any other ship
I didn't say they did. Here's what I said about the writers, director, and film, and the rest after the parenthesis was my speculation:
But, if you want to go with what was said by the writers, director, and in the film itself (that the divergence began with the Narada's entrance to the AltVerse and attack on the Kelvin)...
and if it worked so well, why didn't the Prime Universe go with that design?
IMO, because the Prime Universe Starfleet of the TOS-era never faced the Narada in battle, and never had any motivation to continue building starships of that type to face ships like it. Maybe they were too labor or resource costly to build, maybe they were overpowered like the Defiant, maybe they were not a design philosophy a more peacetime oriented Starfleet wished to pursue further. There are any number of possible reasons.