In the US, too, you are allowed to keep a name if you are using it first and in good faith. You've misunderstood the lawsuit. It was Warlight/Warzone/War.app claiming
Activision shouldn't be allowed to use the dictionary word "warzone" in their game names, not the other way around.
Those Wendy's-type trademark unicorn restaurants are all over the US too:
https://en.wikipedia.org/wiki/Burger_King_(Mattoon,_Illinois)
https://thesubwaypizza.com/In fact, it's the EU system that worked worse for Warzone. In the EU, Activision just filed for and got the EU Trademark for "Warzone" first, because the EU is first-to-file, not first-to-use. They've had it this entire time. It's only in the US where Warzone had the ability to block the trademark because they used it first.