The dragons, or at any rate some of them, were released earlier—at least twice. First in the Nirnaeth Arnoediad:
But even as the vanguard of Maedhros came upon the Orcs, Morgoth loosed his last strength, and Angband was emptied. There came wolves, and wolfriders, and there came Balrogs, and dragons, and Glaurung father of dragons.
(Chapter 20, "Of the Fifth Battle: Nirnaeth Arnoediad")
A second time in the assault on Gondolin:
At last, in the year when Eärendil was seven years old, Morgoth was ready, and he loosed upon Gondolin his Balrogs, and his Orcs, and his wolves; and with them came dragons of the brood of Glaurung, and they were become now many and terrible.
(Chapter 23, "Of Tuor and the Fall of Gondolin")
Glaurung and other dragons of his type were wingless dragons. Fierce as they were, they could only move as fast as they could walk. They would be good for a close-in attack, but not for "projecting power".
It appears that winged dragons were new in the War of Wrath:
Then, seeing that his hosts were overthrown and his power dispersed, Morgoth quailed, and he dared not to come forth himself. But he loosed upon his foes the last desperate assault that he had prepared, and out of the pits of Angband there issued the winged dragons, that had not before been seen; and so sudden and ruinous was the onset of that dreadful fleet that the host of the Valar was driven back, for the coming of the dragons was with great thunder, and lightning, and a tempest of fire.
(Chapter 24, "Of the Voyage of Eärendil and the War of Wrath")
Keeping in mind the long time it took for dragons to develop their full armor (Glaurung first appeared about 200 years after the first rising of the Sun, but his full armor doesn't seem to have developed until the Nirnaeth—that's a quarter millennium of development), it's most reasonable to believe that the winged dragons simply weren't ready for combat until then.