To clarify, are you saying that you think a DM who called for a Strength (Athletics) check to climb a mile-high rope would be acting within the bounds of the specific climbing rules? Or would they be going outside those rules?
Ok, I'm more on the same page here.
I agree with (I think it was Charlequin) that JUST the stakes would not normally be enough to force a check (height above the ground* compared to leaping over a pit with vipers in it).
But I do think that the LENGTH of the climb could certainly factor in. Even with the ability to rest on the knots, this is a method of locomotion which most humans aren't accustomed to performing for any real length of time, and which can tire out the arms pretty quickly depending on technique used.
I think a number of the disagreements here seem to be predicated on differing assumptions about the climb.
Are there indeed lots of hands/footholds on the wall of this ruined tower? What constitutes "sheer"? Earlier in the thread someone seemed to assume that sheer means glass-like. To my mind, "sheer" when we're talking about walls normally just means "there aren't lots of handholds". Your usual masonry wall won't have a bunch of handholds unless the mortar is in terrible disrepair, and even then, those little fingerholds aren't going to suffice for anyone but a good freeclimber/Thief with minimal gear on. Free-climbers don't wear a suit of armor or a backpack; you want to snuggle your belly up to that wall and get as much of your weight against it as you can. Fortification walls are generally built with the intent that people not be able to scale them, after all.
Do most adventurers know good rope climbing technique? Do they routinely know how to properly lock their feet/legs around a rope so as to take the continual strain off their arms? Some of us might assume "yes, of course, all adventurers have a basic level of athletic competence". But I've certainly seen players envision characters (especially wizards) who are unathletic academics, who might have no experience at all with that sort of thing. Even an adventurer who's reasonably athletic might be fine climbing a rope 30' but not have good technique and start to tire after 80', or 200', or some other distance shorter than a mile.
Is time really not a factor AT ALL? Does the Ranger or Fighter have all the time in the world to coach and correct the unathletic character's technique to ensure that he won't screw up the ascent and discover halfway up a long climb that he actually isn't doing the leg lock properly? If time is a factor and the party is trying to avoid possible guard patrols or something, I think hurrying up the rope can be a complicating factor.
*A caveat on my earlier statement, opining that simple height above the ground shouldn't constitute a complication. I am mildly acrophobic. I have good balance and can walk a narrow balance beam at ground level with no difficulty and blindfolded if I need to. Put that same beam 20' in the air and it's suddenly more difficult. Whether there's actually any sway from the trees, or it's just wind, or it's entirely psychological is not always clear in the moment.

I certainly wouldn't impose or assume fear of heights in a PC, but the player may have defined their character that way, and if I wasn't specifically accounting for it, my knee-jerk reaction might have been to think that naturally, being 80' up WOULD make climbing more difficult. So I can understand if that's a factor for some DMs.
All told I'm generally of the mind that "distance/length of a climb can be a sufficient factor to necessitate a check".
That being said, I agree with Flamestrike and others who've opined that making the difficulty too high or requiring multiple checks is usually a terrible idea. And that certain in-fiction realities (being a skilled climber, use of a sliding hitch and/or sizable knots to rest your body weight on and relieve arm strain, having functionally unlimited time) can and should obviate a check entirely.