I see your 'Sky has asthma and that's why he can't run', and I raise you: Sky has Inappropriate Sinus Tachycardia, and that's why he can't run.
Now, before you run away because you've never even heard of IST, hear me out. (If you don't know the condition, it's a dysautonomic heart condition vaguely similar to POTS, just without the low blood pressure.)
We all know Sky's lived above the clouds with minimal oxygen levels his entire life, as have most of his decedents, so it wouldn't make sense for him to have shitty lungs, right? That's where IST comes in. See, it's not related in the slightest to physical health or upbringing, it's just a non-genetic thing that is due to your sinus being a bit funky. It's also not life threatening, so that takes away from Skyloft needing to have non-canonical advanced medical procedures. IST's main symptom is that it severely decreases your stamina by making your heart race and you become extremely out of breath, often meaning you have to stop and wait for a bit for your heart to calm down.
Sound familiar? Yeah, I thought so too.
However, and here's the real seller- unlike asthma, you can just power through and not die. It really, really sucks, but this ability is quite the gamebreaker for me when you think about all of the extreme physical tasks Sky has completed in game, not to mention his swordplay, that if he had exercise induced asthma would've very likely set off an attack. With IST tho, yes it would feel like you were dying, but it is absolutely be possible to just power through your symptoms and have no adverse effects except from your extreme discomfort and fatigue afterwards.
TLDR: Sky has Inappropriate Sinus Tachycardia because it fits him so much better than asthma, as someone who has both asthma and IST. And no, this isn't just me projecting my hyper specific heart condition onto one of my favourite characters, what do you mean....