Quagsire has several advantages ,especially over the Nidos, that could place it in High. The Nidos don't even have access to any Moon Stone in GS prior to credits, but for the sake of the argument let's say you went out of your way to get the Moon Stone in the FLASH chamber of the Ruins of Alph in Crystal.
You can a Moon Stone in G/S immediately after beating Morty (as soon as you can Surf outside of battle). From that point and onwards, you have a huge movepool featuring Thrash/Body Slam/TM-HM normals, Surf, Shadow Ball and three punches, running off a better offence than what Quagsire offers.
-It is one of the better Pokemon to use against Whitney due to its Rock resistance and Mud-Slap, so even a male Quagsire can tango with Miltank. If Miltank decides to get annoying with Milk Drink, you can Tail Whip her until Slam/Headbutt do enough to 2HKO her.
A whole lot of Pokemon have access to Mud-Slap and/or Sand-Attack. Nidorino also gets Leer, Mud-Slap and even a SE Double Kick, and lacking a rock weakness can be seen as an advantage when Rollout takes 3 turns to beat Stomp in damage, is innately less accurate and does not flinch.
-Upon reaching Ecruteak, Quagsire would already have Dig and more importantly Surf, which grants it a much stronger move to use that the Nidos don't have. Granted, its speed means it would have issues with the Ghosts but it can Surf most to death instead of taking double the damage from Curse by Digging (which the Nidos have to do).
Nidos also have the same move right after Morty and will KO some of the same Pokemon without taking damage before that. Nidos do not Dig in this gen so yeah, they're doing nothing against Morty (unless you're one kinky MF and like taking it slowly with Mud-Slap).
-On the sea, while Quagsire's low Speed and lack of Poison-typing means it can get poisoned from time to time, the Nidos have to worry about Bubblebeams, meaning they can't take on everything in the sea.
You need to try kinda hard to get poisoned on the sea, I think. Nidoking will see less Bubblebeams and other attacks due to going first and OHKOing with Thunderpunch or Thrash.
-While Quagsire doesn't resist Chuck's Fighting moves, at least it isn't weak to Water.
In other words, you're reminding us that Quagsire is a water-type. Except it's a slow one without a good special attack stat (and we don't even have speedier specially folks like Golduck up there in High tier for whatever reason).
Neither Nidoroyal nor Quagsire has a good matchup against Poliwrath, but at least Nidoking laughs at Primeape more and can help weaken/finish off 'wrath with Thunderpunch - situational use is still better than trying to solo with Quagsire (which can fail in many ways when you're slow, hit hard and vulnerable to two statuses when going second).
-STAB Earthquake at Level 35??!!??
Like I said before, the expected levels for a full team tackling the E4 is ~L38-39, so Quagsire doesn't spend nearly as much extra time with EQ as anybody else. Bonus points for actually not wanting the TM though.
-Quagsire fares much better against Pryce due to Ice neutrality and also Water STAB.
Doesn't fare better offensively though, besides Surfing Piloswine who is still likely to 2HKO with Blizzard before you grab yours (i.e. "Quagsire is a water-type").
-The only thing Rocket Grunts possess that pose a legitimate threat are Glooms and Vileplumes. While the Nidos do fare better against them in that regard, Quagsire can plow through everyone with Surf & Earthquake alone.
I'm 99% sure Earthquake won't exist during any of the Rocket fights. Nidos will do better everywhere due to being faster (all those Hyper Fangs, Bites and other things common in Rocket fights add up to severe damage faster than you abstain mentioning).
-Guess who can take on Kingdra and who can't?
Oh right, a water-type! Except just about any of them that
resists water has a better Kingdra matchup than Quagsire (who will not outspeed Dragonairs and not deal significant damage to them with Ice Punch due to having a subpar sp. atk stat).
-Quagsire admittedly can't do that much to Will, but it lacks a Psychic weakness so it can OHKO Jynx provided it doesn't get Kissed. Also Exeguttor lacks Grass moves for some reason.
Jynx isn't nearly that easy to OHKO. But yeah, Quagsire is at least taking one Psychic (and not doing a whole lot back) while Nidos have difficulty doing the same, but neither mon is suitable here aside from helping out the rest.
-Keep Quaggy away from Ariados's Giga Drain, otherwise you're fine against Koga.
You forgot to compare him to Nidos here, who do all that + aren't afraid of Ariados (though Venomoth could be taken out with just about anybody else who takes less damage from Psychic). I guess its 4x resistance to poison is worthy mentioning here since we've already begun picking out random details?
-Kill the Onix and Hitmonchan no problem, match of brute force against Bruno's team otherwise.
So Quagsire is an unremarkable slow water-type; meanwhile Nidos resist all the fighting type moves (I'd stay away from Hitmontop though, but anybody can deal with Hitmontop while few are more comfortable going up against Machamp than Nidos are).
-KOes Houndoom and Gengar easily (watch out for Destiny Bond), but stay away from Vileplume duh.
Crunch still really hurts and demands you enter with full health, its damage comparable to Flamethrower on Nidos. Anybody can KO Gengar (though sometimes at its own expense).
-Ice Punch the Dragons duh, but one of the Dragonites carries Blizzard so Nido better be careful. Quaggy KOes Charizard with ease too.
Lance isn't Nidos' best performance at all; yet Quagsire mostly just does what any other water-type would, except worse (provoking Aerodactyl to use Hyper Beam though, which has some merit). It's also borderline 2HKO'd by Zard's Flamethrower while definitely not OHKOing with Surf, so that matchup is hardly what I would call 'easy'.
tl;dr Quagsire is a slow water-type that wishes it were faster and did the things water-types are supposed to do (that is, use water and ice type moves proficiently) more qualitatively. For being so slow, it lacks Graveler's great (on many occassions anyway) typing, so Quagsire users spend more time watching enemies attack, healing Quagsire and hoping there are no status annoyances. It's true that Quagsire doesn't have something explode in its face every battle, but feeling its low speed nearly every fight is painful and for something so slow it could be both offensively and defensively more impressive (remember our assessment of RBY Lax ingame, not to mention Geodude?).