Welcome to the cove and the club.
This is probably one of the biggest problems and of course you are right.
If you look at the
APQP handbook page 3
Effective quality planning involves more than just the quality function.
Most of the tasks are not the responsibility of quality rather engineering or production engineering but the engineers are as we know "not organised enough" and in the end you have to organise the documents etc.
The question is how do you see the APQP team leader. This can be an administrative position of project management or it can involve more. As the team leader you can be responsible for calling the meetings and following up that actions are taken- a type of audit of the process. The actual tasks are performed by the responsible for these actions.
In the end the engineers are the project managers and should really manage their project of which APQP is the automotive method of managing the project.
In the end you need to sell the APQP process as a
tool for designing the process and as such the design should be managed by the function that best represents the design process.
I wish you the best of luck and keep us informed of your actions.