That's called a spillway BTW and the problem with spillways is erosion and the energy of the water. So spillways have to be made of reinforced concrete or similarly strong materials when the local rock isn't hard enough and they have to be shaped to dissipate the energy.

The spillway of the Mosul dam is made of concrete the problem with it appears to be that it allows too high a water level for the dam in its current shape even when fully open.

I'm not sure it would be quite so simple. The article mentions the reason that they can't just open one gate is because the asymmetric draining would be harmful to the dam. It's possible that any sort of adhoc draining would adversely affect the dam. Not to mention quickly building a secure tunnel to handle the water pressure at that depth sounds like no mean feat.