Quickleaf
Legend
I'm really grateful to everyone for their help in trying to balance this out. But, I figured I Should ask the Key Question: can Gorgons get away with keeping the "eyebeams as a subracial trait" idea?
I get where you're coming from when you say that they can't have secondary effects like Paralysis, etc. Even if that poster was mistaken in how often the trait is available. But, these gorgons have being a chaotic, physically/spiritually unstable race written into their fluff. Can the game mechanics support there being gorgons who poison, freeze or incinerate with their gaze?
Absolutely! I think you're on the right track with effects kicking in when you reduce a creature to 0 hit points.
Before, it seemed like you were balancing its gaze attack under the assumption that the extra targets caught in a Dragonborn's breath would roughly equal the conditions imposed by the Gorgon's gaze. As [MENTION=88539]LowKey[/MENTION] eloquently points out, that just isn't the case.
Getting back to that "when you reduce the target to 0 hp" bit, you'll notice that has vastly different implications if being used against a PC vs. being used against a monster. PCs often are being healed up from 0 hp and getting back on their feet, so it is particularly nasty for them. Monsters, on the other hand, rarely do this (regeneration and monsters with healing spells being notable exceptions).
Now, if elsewhere in your homebrew, you were making a bunch of monsters that have ways to come back up from 0 hp, then I'd say your gaze attack could just drop the conditions & keep the "when you reduce the target to 0 hp" bit, and it would be good. However, if that's not the case, I'd suggest tweaking it to be more relevant to how monsters often appear in the game: groups of bad guys that go down hard and fast.
So, you might do something like: "when you reduce the target to 0 hp, another enemy of your choice within 30 feet of it suffers cold/fire/poison damage."
Not saying that gets the feel you want, just trying to illustrate my point.
An alternative would be placing a "death clock" on the monster affected by the gaze attack, something where nothing bad happens immediately, but on a failed save it suffers ongoing damage, and while suffering ongoing damage (ended on a repeated save), if any attack reduces it to X hit points (e.g. 10 hp) while suffering this ongoing damage, the monster is petrified. Alternately you could replace the damage entirely with a slow effect.