Just thought I would put in my input, as I ran into similar issues.
I had a script that would store information into a database, and I needed this to happen every 6 hours.
I was finding that when I scheduled a task, (run once, within a minute, for testing purposes), the script would say that it ran, but looking at the database, it was obvious that the script wasn't running. There was no output or error logs or anything. The script would run when I clicked on "Test Path", so there was something weird happening and I wasn't sure.
I read that if the folder the script was in was password protected (which mine is through .htaccess, for now) there would be problems and that I had to remove the protection.
I also read in this thread that using direct pathing ("../../scriptFolder/script.php") instead of external access ("http://www.myDomain.com/scriptFolder/script.php") might the problem of the script not running, as detailed by the OP.
So just to clarify. I had the same symptoms as the OP (timing fired, but the script wouldn't run except through testing the path), AND I had a script in a .htaccess protected folder. The *only* thing I did was use relative pathing ("../../scriptFolder/script.php") and the script runs perfectly now.
I don't know the specifics of how the scheduler works, but I'm one of those types who is okay seeing something work and not worry about why or how. Perhaps Dave can read this and explain what's happening.