I would say, generally don't count on promises from managers or companies about promotions, raises, or anything else.

Do the highest quality work you can figure out how to do, for the sake of the art. When you're ready for a promotion (like now) if the company doesn't give you one just leave and let another company do it.

It sounds to me like there may be more to learn in your current role though. You could think of this as an opportunity to take ownership and responsibility for the problems of the team, even if they aren't your fault and work on learning how to be a really good lead, even if that isn't the title.

Titles and money are easy enough. The actual bottlenecks are skills, insight, and experience.