When the Brexit vote was in, it so happened that I had entered a short position on SPY a few days prior. I had a thesis that it was overbought and due for a downdraft, although I was not thinking of Brexit specifically. The public overreacted (in my opinion) and SPY crashed, I was able to move fast and make out with like 10% gains over a few days which isn't bad for trading S&P. Too bad it amounted to a $50 gain in my Robinhood account. :)
So yes, average person can absolutely profit from unexpected world events. In fact, for a small investor, truly unexpected events are arguably a golden opportunity, because you don't have as big an information disadvantage vs big traders and institutions. Everyone is equally clueless when a true black swan shows up.
But truly unexpected world events are rare, and macro events generally can be very difficult to interpret in terms of a buy/sell signal. Especially without some very specialized education. And often, setting up an efficient position to take advantage of these requires instruments that are not practical for a small retail trader. So while I say it's possible, I'd also say that it's unlikely to be an effective main trading strategy. It's better to think of it as a sort of betting hobby, and use something more traditional for your actual investments.
As for Carl Icahn, he probably is such a big client that he can dictate all sorts of special arrangements with his broker(s) and do anything he wants. Market hours aren't a huge barrier to trading world events, you can just stick to events that are more conveniently timed than elections, and you can stick to strategies that give you more time to react than a few seconds. That said, if you really want to trade after hours, many brokers allow after hours trading even for smaller accounts. And you could always try trading crypto, which is open 24 hours and definitely reacts to world events (but good luck predicting how).