If you pay attention to DS9 specifically the Dominion War episodes, you see plenty of Miranda's fighting, more then Galaxy's. Why? In war you want a small ship armed relatively speaking to the teeth then a big ship with a large crew capacity.
When you do see the Galaxy's taking part in the action, its always as a tank, sit there and shoot the enemies big ships, as their shield systems and armor is strong enough to withstand the beating while they slowly break apart the enemy.
The Galaxy's 12 phaser arrays, 2 torpedo launchers verses the tiny Miranda with 6 phaser arrays, 2 tubular phasers ("cannons"), and a total of 4 torpedo launchers were used in DS9.
Your also talking about a crew complement of about one thousand verses a maximum of 34 on the Miranda. Let alone the additional construction costs for the Galaxy.
But this is one issue with STO (as a game dynamic not as the game itself), in reality, we should have double the Miranda's and Akira class ships in the PVE's etc, and double the number of fighters in each battle, but then since this is a game, no one would be willing to fly in "lowly" ships to be used as semi-cannon fodder while the bigger ships (Galaxy and Excelsiors) tank.