How the Project Manager participates in Task Management

Each project manager may manage tasks. The following two methods are provided based on their own experience:

Explain the process:

Premise: the customer sends a request and waits for the time to complete the request.

Solution 1:(1) After receiving the requirement, the Project Manager splits the requirement into tasks by role. (2) estimate the completion time of each task. (3) assign tasks to different persons. (4) developers determine whether the time can be completed (5) the project manager adjusts the time (6) Report the time to the customer

Solution 2:(1) After the project manager receives the requirement, it is split into tasks (regardless of the role). (2) estimate the completion time of each task (3) the project member receives the tasks he or she is interested in (4) developers determine whether the time can be completed (5) the project manager adjusts the time (6) Report the time to the customer

Solution 3:(1) After receiving the requirement, the Project Manager splits the task into tasks (regardless of the role). (2) project members receive the tasks they are interested in and estimate the task time (3) project Manager summary time (4) Report the time to the customer

Solution 4:(1) After receiving the requirement, the Project Manager splits the task into tasks (by role), (2) tasks that interest the corresponding role, and estimates the task time (3) project Manager summary time (4) Report the time to the customer

The main difference here is whether the task is split by role, whether the team members receive the task and estimate the time, or whether the project manager divides the task and estimates the time.

I think every solution is not perfect:

Solution 1:

Disadvantage: due to the division of tasks by the Project Manager, some team members may not be willing to perform this task, and the Project Manager may not estimate the completion time.

Advantage: it can control the group's staff's inertia.

Solution 2:

Disadvantage: because no roles are involved, members may be able to get interested, but may not be able to finish the work or be completed more quickly, and the time estimated by the Project Manager may be inaccurate.

Advantage: Members can receive tasks of interest.

Solution 3:

Disadvantages: the project team members may estimate a long or short time, with a large deviation and difficult for the project manager to control the time. The project manager must believe that the team members are not inert.

Advantage: project team members can estimate their actual capabilities. if the initiative is high, it is relatively accurate.

Solution 4:

In case 3, the project manager is split by role, in case some people want to do challenging work, but they actually exceed their own capabilities.

You are welcome to propose a better solution, supplement the advantages and disadvantages of the above solution, or consider which solution is better. Why?