If Shiek is a positive example of specials propelling a character, how many other high tiers are an example of this? (where their specials are technically more valuable long-term?)
In Smash 4:
Diddy Kong; hoo-hah is great and all, but it really wasn't that special when much of the cast had a "hoo-hah" of their own. What made Diddy's so much more impactful? The banana that made getting it all too easy, along with setting everything else up and extreme space controlling capabilities. Monkey Flip too, which aided recovery, got Diddy out of bad spots for free, and is an extremely effective approach move that's very difficult to punish. Diddy is also a character who in patches, has had his standards nerfed significantly while the specials have remained untouched, and what you know, Diddy is still a high tier character and possible top tier, thus being one of the actual examples-in-practise of this.
Sonic: Spin Dash/Charge are the crux of his game, and Spring is another of those "get out of **** for free" cards, that is additionally an incredible recovery move that makes edgeguarding no threat to Sonic. Spin Dash/Charge were additionally so good in Brawl, that they alone managed to make a character with a worse standard moveset than Melee Kirby into a mid tier that won some stuff occasionally.
Zero Suit Samus: Boost Kick is a ridiculously good KO move that give Zamus her terrifying KO power, and Flip Jump is yet another of those "get out of **** for free" cards, while doubling as a good recovery move
and a very powerful KO move/setup. Paralyzer additionally sets up for Zamus' very terrible grab.
Luigi: As I covered before, Smash 4 Luigi's standards have been nerfed very significantly from Brawl's, yet is a significantly more viable character largely because of much better specials (the much better Fireball and more versatile Cyclone) that more effectively cover his grave shortcoming.
Pikachu: Just like in Brawl, without Quick Attack, Thunder, and Thunder Jolt (and Heavy Skull Bash in customs), Pikachu ain't nothing special.
Villager: Villager would be flatout awful if he didn't have the specials he possesses.
Wario: Fart is a game-changer, that matches are won or lost by on if you land it or not. Bike is an incredible move with so many uses and provides Wario with one of the best recoveries in the game. Then Chomp is an amazing command grab that can easily punish many things that would otherwise be difficult to, Without them, all Wario would have is good air mobility.
Rosalina is an example that's hard to say, as Luma is still ridiculously effective regardless of specials, but no Luma shot/Warp would remove her greatest uses of Luma and zoning, and combined with no Gravity, Rosalina would get hard camped by anyone with decent projectiles (which would turn some of her strongest matchups into some of her worst).
Ness' specials aren't too important to him, but PK Thunder is a very useful projectile with a bunch of uses, PK Thunder 2 is a game changer like Fart with how ridiculously early it kills and it's viably landable, and PK Fire is a projectile that's terrible for zoning, but gives extremely good reward relative to the difficulty of landing it and risk of using. Ness with these specials severely nerfed would probably still be a pretty decent character, but they do give him the edge to make him that very powerful high/possible top tier.
Mario is another high tier character whose specials aren't too important, but Cape is a ridiculously good move, and Fireball is a decent projectile that can be zoned with and that Mario's approach would quite a bit weaker without. Then Super Jump Punch is a three frame invincible recovery move, that's effective out of shield and provides a decent finisher to many Mario combos. Mario would still be decent without them, but like Ness they give him that extra edge to compete.
Captain Falcon is really the only example of a commonly percieved high tier who doesn't gain anything substantial from his specials, and he's probably the best character in a hypothetical Smash where specials didn't exist.
Would specials Shiek actually be any better than a pseudo-Brawl Sonic? Simply having good KO moves and decent zoning isn't high-tier material if her entire standard moveset is nerfed to hell, so then it'd be a question of which between the two is better, and how to apply that against the rest of the cast with their standard strong versus special strong characters.
Being able to outzone everyone else on top of great mobility does a ton in itself; see Brawl Toon Link, a borderline high tier whose standards were pretty medicore and had subpar KOing ability, but managed to be pretty successful purely on his zoning alone. Then there's Smash 4 Toon Link, who has what we can consider the "worst standard moveset feasible" and thus be the baseline of what we can consider the baseline "severely nerfed standard moveset", yet he is being consistently ranked among the mid tier over a bunch of other characters, because his zoning is still strong. Then factor in that with "specials Sheik", you're never pressuring her as she's always getting out of everything with Vanish/Bouncing Fish and recovering for free, and can still nab kills at around the 150% mark or less without much difficulty instead of over 200%.
My conclusion was that the more decent options available on a standard set would be more useful to a good chunk of the cast than those cast members having greatly nerfed standards. But, to be fair, across-the-board nerfs in this hypothetical debate are vague. Which is probably why we haven't gone anywhere. We may have totally different concepts of how hard/lightly a character's sets are buffed/nerfed
And with that, I'm going to end the post here instead of responding to the rest with more walls of text, as I feel no satisfactory response can be given, when as you say, we're probably thinking of completely different things.
One question I do have though, is how much experience do you have in competitive Smash as a player and not as a spectator?