You don't diverge from the log unless the Traffic Director comes in and changes it (adding or dropping a spot for instance).
You are about 20 years behind the times. If there even is a local traffic director, they don't go into the studio to change a log. They edit on their own computer and then upload the new log to the digital storage and automation system.
1. When the automation breaks down. Happens more than you might think. Depending on the severity of the crash you are manually re-loading the automation and in some instances, firing each spot individually. It's times like these when you wish you had a triple stack cart deck.
So rare. If the station has a UPS, there should only be the occasional computer issues requiring a reboot and then today's updated systems will take off from whatever point they should be in real time. Traffic will schedule make-goods for any missed spots.
2. While the log may specify a specific placement, sometimes the automation schedules items in a different order. If you're sharp, you catch it and adjust accordingly.
Stopsets are for coffee and bathroom breaks. If the automation accepts the log load at midnight, the spots will never run out of sequence unless production mislabeled one. I have never, ever, heard of automation changing the order of events. The only case where a stopset might get messed up is where station personnel have programed the times for going to the stopset wrong, and you get one stopset being halted to run another stopset.
The paper log holds the board operator accountable for what goes over the air, regardless of how efficient the systems in place are. When you sign each page of the log, you are verifying that everything ran as scheduled. If the wrong spot plays and you check it off as correct, that's on you. Dated copy, a :30 logged but a :60 in the system... these are all things that are the responsibility of the board op to catch. If you check it off or sign off on it, it had better have run correctly
Reconciliation is done now electronically, looking at the "as run" and comparing it with the "as scheduled" data. Many of us reconcile music that way so any songs on slow rotations that did not run the day before can "jump back in the pool" for play the next scheduled day. We use a reconciliation report that flags scheduled but not run songs.
Now I realize that perhaps none of these things have ever happened to you, at stations you've worked at, consulted or heard about. That's OK. These things do happen.
I can see a paper log at some stations, such as smaller markets, where a lot of stuff is done manually and live... such as calling the hospital for birth reports and that kind of thing. But in most cases, it's just superfluous. And technical failures are so infrequent that the losses of spots there is minimal compared to operator error and "wrong cart" mistakes in the "good old days".
Just like someone, somewhere, at some time, paid to have their radio commercial played as the very first commercial in the break, stopset, pod, or whatever you want to call it.
I'm sure it has happened. It's just that I've never sold one or had anyone sell one or had a seller ask about one in 58 years in radio, 53 of which have been in management, generally at multi-station groups and clusters.