Removing a nutrient from your diet (carbs or fat) doesn't make you lose fat if you're consuming a caloric surplus. For losing fat, calorie consumption is the most important factor.
I've known plenty of keto dieters and intermittent fasting proponents who've not been remotely lean despite eating that way for a year or more. It's not that keto doesn't work for fat loss; it's that if you consume a caloric surplus while eating keto you'll gain fat, just like with any eating style.
That said, calories are not the only important factors, especially if you're interested in improving body composition (ratio of muscle to fat).
Many people will say things like insulin sensitivity doesn't matter, only calories. I've even read one evidence-based expert (that I have the utmost respect for) say that insulin won't ever make you fat.
Technically, he's right. Insulin facilitates the entry of ingested nutrients into their respective storage facilities – muscle, liver, and fat cells. Insulin doesn't make you store more nutrients than you ingest. It can't. So, in a way, those who say that are correct: it's the caloric surplus that makes you fat, not the insulin itself.
But if your insulin is elevated above a certain point you won't mobilize (burn) fat as efficiently. If your body has produced a lot of insulin after a high-carb meal, it'll stay elevated for longer. You'll remain inefficient at mobilizing fat for a longer period of time. Insulin's overproduction is what prevents efficient fat loss.
And it affects muscle too. Muscle growth actually benefits from insulin production, especially if your muscle cells are more insulin sensitive than your fat cells. If they are, then you'll be better at partitioning nutrients toward muscle cells.
Insulin isn't always bad. It's important for muscle growth. If all it did was make people fat and it didn't help muscles grow, then bodybuilders wouldn't be injecting it. But they are. This should be a strong sign to keto dieters that the goal of maintaining low insulin levels isn't ideal if you want to build muscle.
Insulin itself is anabolic and anti-catabolic. How? By directly increasing mTOR activation and your muscle's nutrient uptake, and also indirectly by increasing IGF-1 released by the liver.
So even though caloric intake is key in gaining or losing weight (and losing fat/gaining muscle), insulin and insulin sensitivity are also important.