I eat moderate LCHF... Most of what was said on the website you linked. But I will up the fat content as you suggested. I Think I'm a bit afraid of fats...
When I discovered that carbs spike me. I asked the practice nurse why. She told me that carbohydrates are essential for life and that if I didn't eat them, I would be dead in six months. Completely untrue.
Back in 2007, low carb wasn't so mainstream. If you put in 'low carbohydrate' Google decided you meant Atkins. I found an old article in the New York Times by Gary Taubes, entitled 'What If It's All A Big Fat Lie ?' which was eye-opening. Gary Taubes is great on the scientific stuff to the point where his books can be difficult to read ie total concentration needed. Nina Teicholz of Nutrition Coalition, is an associate professor at NYU IIRC. She's great on fats, has really dug into the history of margarine and industrial oils.
If you watch their lectures on YouTube or a few documentaries about fat on Amazon Prime or Diet Doctor, hopefully you will feel less scared.
As a kid. I was given low fat everything, because my father had high cholesterol and high blood pressure. This supposedly perfect low fat diet didn't stop him getting heart disease and neither did going vegetarian.
I had a real fear of fat and it felt like I was jumping off a cliff, deciding to do LCHF. But my BG dropped 13 points to normal levels. I really do understand. But watch some videos and take it slowly.