No i'm not. I'm too lazy to calculate it seems.

That's what it is.

My first impression is that there's a lack of consistency with interaction of the other buttons. "Where am i?"

My proposal is not to introduce a new stopwatch mode but to use the continue mode instead:

Use the time picker button in general to toggle continue mode. If time is set to zero simply activate the start button and put the alarm icon next to the (still deactivated) time display.

This way no time is set to zero as it is now when toggling stopwatch mode, no lack of information when paused if it's countdown or stopwatch, no activation of the time counter bar before starting and no additional mode.