Story where humans artificially accelerate the cultural/scientific status of various planets to create allies against an invasion

Your description sounds a bit like portions of a Doris Lessing sequence I read a while back:

Canopus in Argos

This is Ten Years to Doomsday by Chester Anderson and Michael Kurland

Published in 1964 by Pyramid Books.

Quoted from an Amazon review:

As the book begins, an evil race of aliens threatens the star-spanning Terran Alliance. The aliens’ path to Earth leads through a human-colonized world that seems particularly hapless. As we meet them, the settlers on the planet Lyff seem a quiet people. They have a rigid society which revolves around their king and petty nobility. Even after thousands of years of civilization, the people of Lyff haven’t passed beyond an agrarian lifestyle which barely provides greater than subsistence living.

After their initial reconnaissance, the aliens plan be back in ten years to conquer Lyff and then begin their implacable march through the Terran empire. A stand needs to be taken on this small world, and quickly. But the aliens have astounding technology. How can a tiny planet like Lyff possibly defend itself?

Thankfully the Terrans have a plan: send a team of three scientists to Lyff to help jumpstart the world’s technology. These men start with the introduction of the telegraph but very quickly things begin to take their own momentum and the colonials soon prove to be much more sophisticated than the Terran colonizers expected. What at first seems like indolence or a lack of ambition soon proves to provide a pattern for technological innovation far beyond what anyone could have expected. The arrogant Terrans learn there are more things in heaven and Earth than are dreamt of in their philosophy.

The humans underestimate the Mother religion, and it turns out their holy book was actually about science and they thought it was about ethics. The humans introduction of new scientific concepts made them realize this. There's a scene at the end where the new ships each use a different method to launch, from different rocket fuels, interstellar drives or simply tractoring a star and letting it pull them off the planet. They do defeat the aliens. There is talk at the end of conquering Earth.