I also have a Chief with only frags, and yes it uses ammo in the blink of an eye (and destroys the opponent in roughly the same time span). You can synthesize ammo. The mats needed are hard to mine, but at least back before Odd you could travel to the Crystalline Shards in a day and stock up on high grade raw mats, that you could then trade for ages to the "cheaper" mats you needed for the frag ammo.
Frags can be OP, and my all time favorite ship in the game became a Vulture with frags (rapid fire screening shell). The Vulture is even more agile than the Chief, which I can recommend with frags. You need to be so close that once you fire the frags, the angle they shoot into, will be covered as much as possible by the target. DPS drops extremely fast with distance, but up close they are nasty. In practice I think the frag Vulture I have is more deadly the my frag Chief, simply because the Vulture is so easy to maneuver. Also, with the Vulture you only need to synthesize ammo for two hardpoints, and not six like on the Chief.
At the extreme I also have a T-10 with all frags, called "Broadside" for a reason. It's like one of those giant fish, that wait for their prey, totally still, and then once they become currious enough to come realy close, it's all over in a single shot (small NPC ships). Great fun, and with incendiary frags it's quite a firework.