If you actually read the data sheet instead of pontificating it says of the output capacitor:-
†Optional—improves transient response. Output capacitors in the range
of 1?F to 1000?F of aluminum or tantalum electrolytic are commonly used to provide improved output impedance and rejection of transients.
If your transient response is too good you get an oscillator, which is what happens in actual real life if there is no output capacitor.
The trick is to get a good transient response, achieve a stable circuit, get good rejection of transients and have a low output impedance. These things are mutually exclusive. What you have to do as an engineer is to get a good compromise.
(it’s nevertheless show on p. 18 current 1A regulator w/o output cap, but let pretend it’s a typo for now)
Not a typo but it is in a “Typical applications” section of the data sheet. These are by definition NOT production ready circuits.
It even could drive IC in “overcurrent protection mode”,
Have you ever had any practical experience in electronics? With that sort of nonsense it sounds like you haven’t.