With the proper technique, splitting firewood with an axe in the traditional way--i.e. placing logs vertically atop a stump--can be done safely. And by encircling the log with bungie cords or an old tire, the task can be made relatively efficient, as one does not constantly need to retrieve halves and reposition them on the stump.
But one prerequisite for hassle-free splitting is that the logs need to have been crosscut fairly square, so that the logs sit plumb upon the stump. If the person working the chainsaw wasn't careful, you're in for a headache.
For this situation, or for the inexperienced splitter concerned with having an ax-cident, there is an alternative way to split that you may not have seen before. (Apologies for posting a vid with such a click-bait-y title, the "you've been [doing such-and-such] wrong" is the phrase I hate the most.)
Crazy Russian Hacker didn't invent this method, of course. As one commenter pointed out, "[It's] amazing how the old ways are forgotten. My grandfather split wood this way, he learned it from his grandfather, who learned it from his father a Swede lumberjack."
I don't doubt that someone will do this, leave too much overhang on the log to be split and send it flying back towards them, but you've got to let Darwinism take its course. I'm going to give this a shot out at the woodpile.