Climbing has some inherent danger, obviously, but with a bit of knowledge and staying within some reasonable safety limits, the likelihood of dying is nowhere near BASE. I think part of that is that, at least amongst the climbing community, the celebrities aren't wholly people pushing the limits with danger. Sure, we have Alex Honnold, but we also have Tommy Caldwell, Chris Sharma, Adam Ondra, Alex Puccio, and so on, climbers who are pushing the limit in difficulty but not really danger. Perhaps that's because "difficulty" and "danger" are not inexorably linked with climbing. You can have a close-bolted route in good rock with modern bolts that's not particularly dangerous that's still really, really hard. La Dura Dura is the hardest sport climb in the world. I probably wouldn't kill myself on it, even if it's way above my pay grade (assuming I could even reach the first bolt). I don't know that you can do the same with BASE.