Damage building is, by far, one of the most important aspects. You mostly use standard attacks unless you're playing a few key specific characters. I'm not trying to undersell specials here, you're right in what they give- with recovery being (on average) the most useful aspect a special is going to give, but if Shiek's specials were made bad, she'd be more likely to still be top tier than if her standards were similarly made bad. That's my point. This is going to apply to most characters.
There's a lot more that goes into the game than damage building. Spacing/zoning opponents out to get safe damage and setup opportunities to build damage at all, avoiding damage so you keep the lead and don't get killed, finishing off stocks. etc. If pure damage building was what mattered most, Brawl Sonic would have been one of the best characters instead of a mediocre one, and Brawl Snake wouldn't have been that good.
For your example, if you made Needles, Vanish, and Bouncing Fish bad, Sheik would absolutely not be top tier anymore. Those are by far the most important moves in her moveset, and are the biggest contributors to why she has been such an OP character. Needles let her zone out everyone despite being a rushdown character and are used for important kill setups/gimps, Vanish gives her an absolutely free recovery and a really effective KO move to finish off stocks early, and Bouncing Fish recovers, finishes combos, kills, and gets Sheik out of bad situations for free.
The only special that explicitly holds Falcon back is his poor recovery, but giving him a decent one doesn't change the fact that he relies on momentum. If you're in a position where you're required to recover as Falcon, you still have an uphill battle to actually start gaining momentum. Otherwise, you're combo food. Those are Falcon's big weaknesses, and they don't simply stem from "bad specials".
That bad recovery is Falcon's biggest flaw and is the main thing holding him back, especially in Smash 4 where bad recoveries aren't the norm. And Falcon being "combo food" comes from, besides falling speed, him lacking specials that gives him protection when sent into the air. Time to bring up Smash 4 Sheik, she's a fast faller like Falcon who lacks aerials with disjointed hitboxes, but why doesn't she get hit anywhere near as hard when sent into the air? Because of Vanish and Bouncing Fish getting her out of such situations. If you made Falcon Dive a good recovery move somehow, and turn aerial Falcon kick into a move that could more reliably get Falcon back to neutral from the air, Smash 4 Falcon would definitely become a top tier.
Ice Climbers are held back by their linearity. They rely on Wobbling to compete at a top level, meaning they're predictable. There's a set strategy for countering them: Separate Nana and Popo, and that's their downfall at a top level.
That linearity is a result of specials that give them little additional effective options and that fail to cover the holes in their offenses and defenses.
it's really the whole moveset. You have your examples above, but my point is that if you took existing characters and nerfed their specials and left them with their standards, they're more likely to do well at top level play than vice versa. Diddy was an extreme example that wouldn't reflect most of the cast in this game.
Again you're really underselling the impacts of specials to a character's success. I covered Sheik. With Zamus, if you took away Flip Kick and Paralyzer, amd removed all of Boost Kick's KO power, Zamus would plummet through the tier list. If you took away Sonic's Spin Dash shenanigans and combos, and took away the invincibility and distance of Spring (as well as the not giving helplessness feature), Sonic would be outright terrible. If you made Luigi's fireball a lot laggier, and removed Cyclone's mobility and height gaining feature, Luigi would become pretty damn bad. You take away Diddy's banana and Monkey Flip, he would end up a pretty mediocre character.
With Rosalina it's difficult to discern. But you take away Luma shot, she wouldn't be able to space Luma out anymore, one of the most pivotal aspects of her character. And Gravity is the major reason she is such a hard matchup for characters who rely on projectiles (and Ness/Lucas), so losing it would hurt her a lot more in these matchups than losing any single standard would.
She's a light and floaty character with one reliable aerial. That's all that really needs to be said. A better ability to camp isn't going to maker her viable when she's slow, lacks good mobility, and has nothing to back a theoretically good Din's Fire up except a couple of good tilts and an okay Nair.
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She's a light and floaty character with one reliable aerial. That's all that really needs to be said. "
Yeah no, this kind of attitude is what leads to inadequate information being given and people saying what amounts to bull**** (e.g. Mii Swordfighter is totally bottom 5 still!)
If Din's Fire was a good projectile that could be used to effectively zone with, it would mean
opponents would have to approach Zelda. Which means Zelda could actually utilise her moveset defensively like it's tailored to. It would mean her poor mobility is a minor annoyance instead of being crippling (poor mobility hurts you a lot more when you actually need to approach and chase people; see Brawl Olimar for why poor mobility doesn't matter when you can camp everyone out and hit hard when they get in). Suddenly Lightning Kick won't be so bad, when you can afford to throw them out, and hard hitting moves that don't really combo would become a lot more meaningful when your opponent is soaking up more passive damage and being more pressured.
You really need to learn your Zelda stuff, and how moves intertwine into a character's design, before spouting off stuff like this.
Would Zelda become a good character solely because of Din's Fire being good? Highly doubtful, but such a buff would go a very long way to making Zelda a usable mid tier, much moreso than any non-ridiculous buffs to her tilts and aerials would.