I'm sorry, but you are trying to solve the wrong problem. Your team isn't delivering much because your management has no expectations from them to do otherwise.
There is a saying in my country that literally translated means "As you teach them so you have them". Management taught this team that nothing happens if they don't deliver much, taught them that nothing happens if goals are missed, taught them that it's OK not to finish work. As you teach them so you have them.
I've seen this in every company I ever worked in. There are some high profile projects with strict deadlines, everyone works to the best of their abilities, there are stakeholder meetings, there are demos of the product being done, goals are set, work is managed to reach those goals, if deadlines are missed people notice and scramble to reorganize their work, etc.
And there are also some projects where everyone is just chilling. Management is so busy with the high profile projects that these other projects just fall through the cracks. If people keep quiet and avoid drawing attention to themselves, they can just fly under the radar indefinitely. Well, maybe not indefinitely, but quite some time. When management eventually notices, these projects get cancelled or hard deadlines are set or a lot of pressure is applied on the people to finish the work 'cause "what have you been doing all this time?". People leave or make a mess of things because the deadlines are unrealistic on the new schedule (would have been fine on the old schedule if people would have been working, but they weren't). So eventually, the projects get cancelled too.
Your problem isn't with the team, it's with management. Management needs to express their expectations, mention the consequences for not meeting them, collaborate with the team and get involved in actually managing this project. You can't impose hard deadlines (or simulate hard deadlines, as you say) but then keep looking the other way as things continue unfolding in the same manner.
Management needs to do its job from the start of the project, not just when there is a blip on the radar and then wonder "Where did this come from?". Spread your attention wider than the team. Everyone needs to get their s#it together, not just the team on the project.