The fairlead should limit the rope to within the drum sides, i.e. it would only go to the sides if bunched up.
Given the damage, I presume you had no one looking to see how the drum was loading up as you winched... I use synthetic so am not particularly afraid to stand to the side and keep an eye on the drum loading up as I winch in (electric).
With steel rope you don't need to grease it. I presume it is in good condition.... It should load on the drum reasonably well under tension. If the log wasn't heavy enough to keep tension, i.e. it rolled along etc, then the rope could have too much slack and bunch up or tangle etc...
Winching for a long length without stopping to check how the rope is laying on the drum as it spools in isn't a good idea. I'd typically winch 10' at a time and then look at the drum when I ran steel rope. Also put a winch blanket on the steel rope in case it snaps etc...
Anyhow, sounds like insufficient tension was one issue (how big/heavy was the log) and also not checking every so often how the drum was loading up was the other.
cheers,
george.