Now that me, the NMOS newbie who had not read any introductory text book on this subject before, has spent some 20 hours reading Jaeger's book 4th Ed, has gained adequate prerequisite to understand the OP's question.

The OP's question is in two parts:

(1) (a) when the gate is shorted to the drain on an NMOS, Vds = Vgs.

(b) So Vds >= Vds - Vtn, making the MOSFET always in saturation.

Now Let me start with (1) (a) which turn is in two parts:

(1)(a)(i) Short Gate to Drain,

(1)(a)(ii) Vds = Vgs

(i) obviously causes (ii). But why do this shorting?

It is only recently that in another Q&A (Note 1), when someone asked about how to use 2N2222 as a temperature sensor, most answers suggest to short Collector to Base, so converting the BJT to a diode (BE junction). And since BE junction voltage is dependent to temperature, so can be used as a temperature sensor.

So I was misled by the above 2N2222 CB shorting to make a BE diode solution, and wrongly thought that the NMOS (Eg URL540N) should be similar. But this is a severe careless mistake I made, leading me into a dark alley.