I've just faced similar problem today, it was an simple application, and the task can't start.
I got the green line, that the task is started, but nothing happened.
I create a new task, that always give skipped status.
Restarting the services was the solution.
I hope your problem's solution is just as simple.
My case is more complicated (and I've tried your solution as well).
The "Skipped" status was appearing randomly (I've had 10 tasks starting at once).
Re-arranging these tasks to start in two groups (5 and 5), 5 minutes apart fixed the problem.
Qlik Support confirmed it as a case, and they are "investigating it".
I have some other issues where Qlik is not handling multiple simultaneous tasks well, but it's more about the system resources issues....
Will keep updating this post if I will get some resolution from support..
QS should reload applications regardless in they "were already reloaded" or not.
Some of my tasks are scheduled to be executed every 15 mins and they are executed every time...
I am assuming the task's schedule was heavy at some point in your case which caused the task to go into the "Skipped" status.