Source: the signs outside Northern California open spaces (and the newspaper)
It is estimated that 50 million Americans go hiking each year, and of them 25,000 will die out on the trail. Hiking has some inherent risks. Cougar attacks should only be slightly higher than "swarmed to death by killer bees" on the list of reasons you don't go hiking or trail running.
Key phrase: if you can. I'm a runner too, and I run year 'round in a suburb that has all of the problems others here have mentioned. It is possible, and I've even written guides on how to do it safely. It just takes more effort. In particular, wearing headphones is even less advisable in winter than it is the rest of the year. You need to hear when vehicles are approaching, even if they're EVs and all you're getting is the tire/wind noise. It's certainly not for everyone.
For those who don't want to make the effort, I recommend cross-country skiing or snowshoes. Work with what you have.