Sorry, I'm not hearing what harm it does other than it's not to your taste. And you're not at my table, so you don't get a vote.
Of course, at your own table you can decide what is cool and rule accordingly. But all I'm trying to show is why if you are at my table, rule of cool doesn't resolve the real conflict here.
I think what you are asking when you ask, "I'm not hearing what harm it does other than it's not to your taste", is something like, "If the player gains no advantage from it, why can't you add it is color as part of the adjudication of success?" And sure, I could arbitrarily adjust the difficulty of sledding down stairs to be the same as the difficulty of running down stairs simply because I want the player to succeed here. Or I could even simply add the sledding down stairs part into the narration as part of the resolution narration, even if the character didn't explicitly propose it.
But I don't either one at my table, and ironically I'll cite the Rule of Cool as to why. Because I don't think it's cool. I'm not the part of the audience cheering at that scene. I'm not the part of the audience laughing at that scene. I'm the part of the audience groaning at that scene because it isn't cool. Sorry that my aesthetics aren't the same as yours, but as you noted, you aren't at my table.
To me, that doesn't look like punishing the player for trying the stunt, it is punishing the player for trying the stunt. Especially depending on how high you set the difficulty.
No, it isn't. It may look like it, but it isn't. This is very important. That the difficulty of sledding down stairs is higher than the difficulty of running down stairs isn't an arbitrary decision, but a decision based not just on how I want this scene to play out, but how I want all my stories to play out. That is to say in a very real sense I'm not deciding it then at the point that it came up, but I pre-decided it before we ever started play. It is part of my demographics, logic, and physics of the world that I'll appeal to to figure out what I should rule when I don't have something already written down.
UPDATE: Let me add to that by talking about how I would decide how hard it is to slide down stairs on a shield. So confronted by this question, my thoughts would probably first turn to tic tock and stupid people stunts or those "people are awesome" videos where some person has practiced some trick over and over and then records his success. Sliding down stairs on something is on the lower end of that that doesn't require a world class athlete, but it is still something you probably don't want to try to do until you are pretty proficient "board slider". So I'd probably think something like, "Well, this is probably something a second or third level character with decent DEX has a decent chance of pulling off without breaking their face.", and I'd set the difficulty accordingly. What I never would be doing was going, "What do I want to have the outcome be?"
UPDATE #2: Let me further add that if you are at player at my table, and you think it is cool to slide down shields to no purpose, and I don't, then the rules provide for you to gain that narrative power that you want and override my taste preferences. If you are playing a medium to high level character with high DEX and you take something like my Poetry in Motion feat, you can have your character showboat sliding down shields til your heart is content while I the GM roll my eyes and laugh "unable to do anything to stop you" because you are using my rules "against me". (Rules that in fact I created to provide players the ability to do that sort of thing if they really want to.) But notably, that isn't relying on "Rule of Cool".
If Legolas needs a natural 20 to do this and ends up face planting and Lady Delia is never seen again, the player is going to learn there's no point in trying stunts. That way lies players who only go down stairs with a safety line clipped onto the banister and poking every step with a 10' pole for five minutes before stepping down.
See at this point it reads like you are spittle flinging rage as you type at your computer.
And I don't really understand why shield-sliding down stairs is bad but shield-sliding down a mountainside is acceptable.
Yes, but your lack of understanding is your problem and not mine. The fact that you don't understand shouldn't provoke you to assume I don't have a sound reason for my preferences. It should provoke you to question how much understanding you actually have.
The reason why shield-sliding down stair is bad but shield-sliding down a mountain side is awesome is in real life we do one of those things because it actually makes sense to do it. We don't normally slide on a board down stairs for any practical reason because sliding on a board down stairs is impractical and generally accomplishes not a whole lot. But we do normally slide on a mountain on a board, and we call those boards sleds and skis and snowboards, because at the root of that sliding down a mountain on a "board" is in fact a very practical thing to do. Real world militaries operating in arctic conditions still use and train with "boards" on ice and now precisely because they can then move with greater ease than those that aren't proficient "board sliders".
So in one case the character is leaning into the fictional setting and engaging with it in a practical manner that conforms to the setting tropes. And in the other case the player is leaning out of the fictional setting, disengaging from it, doing nothing practical, and ignoring the setting tropes. Which do you imagine I think is awesome?