The assertion that LCHF is healthy, is a controversial one, with most health professionals recommending a balanced diet, and a few outliers recommending LCHF and asserting that the health profession is uneducated or unscientific. There are also a few outliers at the other end recommending a low-fat diet, though that was more popular a few decades ago.
The conservation of energy is a bitch if you're trying to lose weight. Calories are a measure of the energy in food, and that energy has to go somewhere. You can poop it out undigested, but that's a tiny part of the calories you eat; you can burn it through metabolism; or you can store it as fat.
Increasing exercise is the healthy way to burn more calories, and has other benefits besides. There are chemical ways to increase metabolism but they are fraught with very nasty side effects and are addictive. At the very least, when you stop taking them the effect is reversed, your metabolism drops way down, and you gain back all the weight you lost.
I think the controversial bit is that by getting into ketosis you enter this magical weight-losing paradigm in which you can eat all the steak you want or something and not lose weight. It's not *terribly* controversial that if you go to extreme measures to restrict your calories, you'll lose weight if you consume less than you use up. And anecdotally I've known a few guys who lost a *lot* of weight on Atkins. I do think that it has many of the same issues that other diets have - namely, that if you just use it to lose weight and then go off of it, you're probably going to gain that weight right back - but in the sense that it gets people to eat less, sure, it seems to work pretty well.
Also, it is all but impossible to lose weight only or primarily by exercising more. If you're very overweight and you can't work out strenuously for more than let's say half an hour, you might only burn 2-300 calories compared to your resting metabolism. And to make matters worse, it's been shown time and time again that people who work out, especially those who are kind of new to working out, tend to cut corners at other things (for instance, taking the elevator instead of walking up the stairs) so that a lot of the benefit of the exercise is mitigated. I mean, you *also* can't just expect to lose weight by dropping to 1000 calories a day - all a starvation diet is going to do is mess up your metabolism - but at the very least you need to do *both* the exercise and the diet if you want to lose weight, and I guess if you really, really wanted to dump one of them (which you shouldn't IMO). The ideal way, I guess, is to get yourself to a point that's close enough to stasis that you don't feel hungry all the time (or for that matter send your body crashing into starvation mode) but maybe still just a bit below and then just let the weight come off over time. We don't really live in a culture that approves of that, though, so it's tough.
Personally, I'm generally a fan of the idea that people should be more active just because it allows you to do more things in general. This applies whether you're morbidly obese or if you're HWP. Finding stuff that burns calories and elevates your heart rate is sometimes not easy and not fun but I think that most people would benefit from doing more of it.