Actually, they are correct. The only nuance I would add is that, from reading the scientific literature, we are not exclusively in fat or carb burning mode, it's usually a mix of the two. The mix depends on how well your body is adapted to fat oxidation, your blood sugar/insulin levels, etc.

The podcast you linked is very interesting but doesn't necessarily refute the parent comment? You can improve fat burning in a multitude of ways but I believe the best ways are to avoid carbs/sugar and do quite a bit of training below your aerobic threshold. Most people have such a low aerobic threshold that any exercise they do is above the aerobic threshold, so they won't "burn fat" by exercising, instead they use liver/muscle glycogen.

I've always been a runner but a few years ago discovered the low carb diet. I eat less than 50g carbs a day now. I'm very well adapted to be a fat burner and have very little body fat compared to most people < 10%. I did a metabolic test recently and at my aerobic threshold I was burning 2g of fat per minute (which is alot), at an aerobic threshold HR of 158 bpm (lactate threshold at 177bpm), which enables me to run at 6.30min/mile pace and I can keep that up for hours without bonking.

If I eat a piece of bread and go running, my aerobic system never really kicks into gear because blood sugar/insulin levels are too high and the above doesn't hold.