I haven't been able to keep up with this forum but I just heard about the taxi strike that went on for several days.
The link provided in the wiki above that directs you to the transportation section of "Wanted in Rome" has some recent articles on February's taxi strike - which was followed by a Public Transportation strike. (I'm posting with my phone otherwise I'd provide links here.)
All this to say, I continue to feel the most comfortable with using a car service for when I have to make a TATL flight out of FCO. And it's what I continue to recommend to family, friends and businesses associates.
The taxi strike lasted six days and is over, the public transportation strike that added trains and buses was for only four hours. The Leonardo Express doesn't go on strike.
Private cars and Uber are not fool proof solutions. This strike was a protest against Uber and private car services.
If you look at the videos below I doubt that too many of the drivers they were striking against would risk being out in their cars. There were even reports of a gun being pulled on an Uber driver who attempted to pick up a fare during the strike. The streets were blocked, so you would not get anywhere, private driver or not. Scroll to the bottom of the first link and look at the video to see what I mean. Also below are some of the images I think you were referring to.
http://italiapost.it/protesta-taxi-r...-polizia-video
https://www.rt.com/news/378111-clashes-italy-uber-taxi/
http://www.wantedinrome.com/news/rom...e-intensifies/
http://www.wantedinrome.com/news/rom...lent-protests/
Strikes happen, but they are usually much more minor than this, and you always get a few days warning which gives you time to make plans. If I arrived to the airport with a reservation for a private car and taxi drivers were rioting in the streets against them, I'd have doubts about whether my driver was going to show up. I wouldn't have put my faith on a private car driver wanting to drive through taxi drivers rioting against them these last six days; I'd still have alternatives planned out.
There are times when a private car to the airport is better, such as getting to FCO for a very early flight, where there would be no cars at 5 AM at the taxi stand. You'd have to call one and they run the meter from the start of the call, not when they pick you up, although a taxi is more reliable because you have the whole fleet, and are not relying on one person they gave the reservation to hearing their alarm and getting out of bed.
From FCO to the city I prefer a taxi, because again you're not depending on one person. I've had car services stand me up or make me wait, or have to walk for 15 minutes to a parking lot where they left their car. There is not a private car service stand, like taxis have. There's always a line of taxis waiting just outside the door, whereas the reliability of car services can be hit or miss.