The turn procedure is used during an engine out situation when performance is reduced, and is designed to provide terrain clearance by routing around obstacles. It is designed to move the aircraft out of a congested area quickly, and simplifies the departure to give the crew the chance to level, accelerate, and begin handling the problem.
ATC doesn't have our turn procedure. Only we do. We notify ATC as soon as possible what we're doing. "XXX with engine failure, turning left 180, we will get back to you a shortly."
Most places where we have them, meeting climb gradient criteria isn't the issue. It possibly could be, but we're not going to keep climbing. We have a standard level off altitude of 800' above the departure elevation, which is modified by virtue of the program used to calculate our departure performance criteria. While the aircraft could continue to climb, the idea is to get cleaned up and accelerate to a safe airspeed as quickly as possible, enabling better climb performance sooner. The turn procedure is what makes it possible.
When we print up the TOLD card with our departure performance data, one box will tell us if a turn procedure applies. We don't rely just on that, of course (we do look in the book, or on the EFB), but it's all spelled out right on the card, including the 3-engine level off altitude.