You're conflating the right to do something with whether it is advisable to do something.
Sure, you are ~allowed~ to begin an immediate descent in an emergency, but it is not a good idea considering from the pilot's perspective, the bang is most likely an engine going out and altitude is always your friend in this condition.
In an emergency, caring about ATC is literally your lowest priority, in every case. It is ATC's job to notice a plane no longer under their control and route other aircraft safely around it. Your entire job is to do everything you can to prevent the death of anyone onboard, and playing ham radio is rarely the best way to do that.
