Yes of course! My entire childhood was spent running around in the woods. Bears and mountain lions are rare and most of them are more scared of you than the other way around. No more than a small handful of people are attacked by those across the entirety of North America, the last time a child was killed by a mountain lion was about 25 years ago. There were only two or three fatal mountain lion attacks against kids in the 90s, two in the 80s, and three in the 70s.. there are less than 20 of these kind of incidents in the entire 20th century across all of North America. Why the hell would you bring up such an unlikely risk? And fatal bear attacks are only marginally more common, they're still incredibly rare.
Your kids are many thousands of times more likely die from choking on food than by getting attacked by an animal in the woods, do you ban grapes in your house? How did you become so cowardly? Forbidding kids from playing in the woods is absolutely pathetic, reading your comment left me with a feeling of visceral disgust.