You can perfectly use both if you know what you are doing. They don't mutually exclude. Although day to day keto is a lot of fat and meat. You may want to favor the vegetable part of keto after a long fast for some time.
People who do keto do get a majority of their calories from fat, but a majority of long term ketors tend to eat mostly low NET carb vegetables by volume. For example, you could eat a whole small cabbage in a day and not fall outside macros that would allow for a ketogenic state.
>Fasting have other benefits than keto: it activates your immune system, increase you hormone levels, allows many organs to rest and clean themself.
I have seen studies that show many of these as the same benifits of keto. For example, NAFLD resolves within only slightly longer on keto than by fasting. Keto is pretty effective at resolving PCOS for example.
Now, it maybe the case that fasting is more actually effective, but keto unlike fasting is sustainable as a long term lifestyle (conserves LBM etc).
As you pointed out, intermittent fasting and keto are completely compatible.
IF this is correct, it means keto and fasting both accomplish the same thing in terms of life extension, only with keto you get to eat bacon, steak and cabbage, spinach, brussel sprouts, etc rather than starve and be miserable. :D