We start to use timers and I think that a manager should be able to se a task as a countdown timer. This means that the estimated work hours should be used as clock ticking and the owner should accept that this is the time to complete his work.

This will increase productivity dramatically and also wont let users forget other tasks in the pipeline. I am constantly spending more time on tasks to optimize the result but an hour MORE on a task means an hour LESS on another.

A perfect result is nice but no Result at all is catastrophic. When we steal time from other important tasks this happens. A countdown timer will fix this problem. If you add pop up notifications also would increase productivity. You have 1 more hour to complete your task. You have 30 more minutes to complete your task.
But if you do this you should give us the way to make it happen in specific tasks because it will also be stressful for the employee having all day a bomb ready to explode in his hands. You also have to give the ability to the user to set as many reminders as he wants. I would choose for example in 2 hour task reminders like

One hour

30 minutes

10 minutes