As for Cleanse removing GW, GW is essentially a quasi-CC in that it negates lifesteal, spellvamp, and other combat healing which some people rely on.
Counting Grievous Wounds as quasi-CC is a huge stretch - one which essentially counts all debuffs as quasi-CC.
Main difference between QSS and Cleanse is all debuffs vs. CC as well. Cleanse removing the DoT of Ignite is the only exception to that I'm aware of (it doesn't even remove the %DR of Exhaust). DoT's of Teemo, Darius, Cassi, and more are unaffected by Cleanse, same deal with armor-shred debuffs, %damage reduction debuffs, and so on from the likes of Urgot, Malphite, and Black Cleaver.
The big problem with Cleanse isn't that it's not amazing vs. Ignite though - it's that there are quite a few CC effects it simply doesn't effect. Knock-ups, Knock-backs, Flings, pulls, suppression, localized slows.
A solid 1/3 of the champions in the game have something that counts as one of those "true damage" types of CC, which makes Cleanse pretty useless against actual CC heavy champions. Out of Alistar, Singed, and Blitzcrank - all you hit is Blitzcrank's 0.5 second silence.
Which makes Cleanse extremely unreliable and unappealing at doing what it's advertised as doing.
Also - only Suppression and Silence actually disable the capacity to cast summoner spells that don't fall into the 'mobility' category, allowing Barrier and Heal to be cast in pretty much every situation Cleanse could be (barring silence). As it's entirely possible to run into champions like Alistar with CC completely unaffected by Cleanse/Tenacity, or it's entirely possible to run into champions like Corki, Zed, Ezreal, and Nidalee that have no CC or very weak CC - Heal and Barrier are just way more useful in most situations.
Iceborne Gauntlet's slow is unaffected by Cleanse or Tenacity as well, so there's even Cleanse/Teancity-immune CC in items.
With Cleanse - you almost have to play Draft to know if it's a worthwhile spell to run. I still see it occasionally on Vayne against the right teams, but overall - it's extremely niche.
Dominion is the only map I've really been happy with running it in blind pick, as Jax, Pantheon, and Teemo are pretty reliable opponents to run into - and removing their CC can offer a significant edge against them. With a 5v5 brawl you can also easily focus that 1 guy you can actually cleanse CC away from, where in Rift you're generally matched with 1 opponent and can't go headhunting for any opponent.
is glad you don't bring 
It's a cool idea, but the mechanical reason why cleanse doesn't remove Grievous Wounds is because cleanse only works on crowd control effects, and grievous wounds does not come with any CC.
Quicksilver Sash on the other hand, is designed to work with all Crowd Control Effects, Damage Over Time and Debuffs. It also works on ignite and exhaust.
