The mechanics themselves? As in the physical rolling the dice and whatnot? Sure, that's not diegetic.
Nor are many of the words that the players speak, like "I [as my PC] turn around." No in the fiction hears those words spoken.
As I posted not far upthread, this is a recurring source of confusion and tension in RPGing, when the GM asks the player "Did you really say that?"
But, what those rolls actually represent can be. If the roll provides information about how a result was achieved, then that information is diegetic - it is known and knowable within the game world.
The stuff that is represented might be part of the fiction, sure. The act of representation typically isn't, though - as per the example of the player saying, as their PC, "I turn around".
What is part of the fiction is
the turning around, not the act of saying "I turn around".
That's why the mechanics can be said to be diegetic. I suppose the more accurate descripition is that diegetic mechanics provide diegetic narratives - not that the mechanics themselves are diegetic.
Here I think we part ways. Because I agree that, speaking accurately, the mechanics are not diegetic. But isn't "diegetic narrative" a tautology? Or very close to one at least.
They are no more diegetic than the script that the actors are reading from, but, when discussing diegesis, we don't really have to worry about that. That's not the issue. Yes, we, the audience, know that there is a script and a director and all of that. Fair enough. Even though none of the actual creation of a movie or novel is diegetic, it produces a diegetic result.
I think I agree with this. But then I want to restate your points about the climb check - which basically I agree with - like this: the climb check isn't representational. And so it does not introduce elements into the narrative (that is, into the diegesis). As per my post somewhere not too far upthread, it's like the film cutting from the character getting ready to climb, to a scene where the sweating, red-faced character pulls themself over the top of the cliff (success) or to a scene of the character tumbling down the cliff to their doom (failure).
This is why I liked your map and puzzle square examples: because these are ones where we don't even need to talk about representation. What the players are doing (poring over the map or the puzzle) is exactly what the characters are doing.
But if we talk about representation, we at least need the mechanics to map to descriptions of processes or events. The Rolemaster Move/Manoeuvre table does a bit of this - eg, here are some of the failure entries, in increasing degree of seriousness:
Fail to act
Freeze for 2 rounds
Fall. Sprain ankle. You are -30. +15 hits
Fall. Knock yourself out. You are out for 30 rounds. +10 hits
This still doesn't tell us exactly what went wrong for the character, but it does provide descriptions of whether they choked at the outset, or tumbled down the cliff.
TL;DR - I don't think we are disagreeing that much over the basic points at issue.