Imaculata
Hero
For reference, I'm with you to the extent that I wouldn't call for a check for climbing an 80' knotted rope. (I would call for a check for, e.g., a 1000' rope, and the implication of a sharp cut-off at some distance X doesn't bother me, since I'm never going to have an X' and an X'+1" rope next to each other in the same game.)
That seems incredibly arbitrary to me. 5e is pretty clear that climbing is just a fraction of your movement, and not a complication in and of itself. It's right there in the rules. I know 5e isn't as wordy on the rules as 3.x, but this seems like basic reading comprehension to me.
Now you can houserule it all you like of course, but the discussion is about the rules provided.
then if a particular DM considers a potentially lethal fall to be such a complication, then, by my reading, the rules encourage that DM to call for a Strength (Athletics) check.
A potentially lethal fall is a failure state, not a complication. A complication is something that 'complicates' the climbing (makes it harder to climb the rope).
but the rules leave the question of what qualifies as a complication up to the DM.
Yes and no. Yes, the DM can decide what is enough of a complication to affect the climbing, but no, the fact that you could fall is not a complication, nor is distance climbed.
So if, for example, @6ENow! rules that the stress of a potentially lethal fall qualifies as such a complication, I think they're following the rules to call for a Strength (Athletics) check.
Unless the PC is specifically afraid of heights, I don't think it does. Heck, you could have PC's make checks for the stress of swimming across a pond because they might get bitten by a crocodile. I don't think that is how checks work. Fear of failing a task should not contribute to the difficulty of the task, unless the character has some kind of phobia.