Why does no one take Ignite on ARAM?

There are better options. Right now, Mark/Dash is the dominant one. It's loaded with utility on a low cooldown. If you're just concerned with damage, it's still got a higher potential output. You can do more true damage than Ignite at a longer range in the same amount of time it takes for one Ignite cooldown, even with just the 25% Mark damage. There's no Innate Grievous Wounds effect, but that's not a huge issue due to nerfed team heals.

Ignite is primarily used to give you the advantage in a 1v1 or to finish off a kill. In aram there is no way to back so being able to finish someone off is much less important and the constant teamfighting renders the 1v1 advantage negligible. Most people prefer the survivability either heal or barrier because they are stronger for teamfighting and flash because of its general usefulness.

Because of the range, and the fact it's a constant 5v5 teamfight.

Exhaust, heal, and other spells have better utility, and sometimes utility is better than damage due to benefiting the whole team.

Ignite is normally for the finish, but in ARAM they're not going anywhere.