It is never stated that Sheogorath in Skyrim is the CoC in Oblivion, though Sheo implies he was present at certain events of the game

Except that is pretty much means that Sheogorath is, in fact, the Champion of Cyrodiil. Seeing as how the only Daedric Prince present at that event was Mehrunes Dagon. So Sheogorath could have only been there in his previous incarnation as the Champion.

he starts Sheogorath's Quest.

There are easily other reasons imaginable for what Sheogorath says. The mantler could have been another person present at those events. Sheogorath could be speaking metaphorically, he must have been 'present' everywhere in those mad times. Personally for the reason of things not becoming extremely silly, I tend to imagine that the only thing any of the games heroes did was the main quest. The rest may have been done, but possibly by other people.

Plus, the only thing that made the player in Oblivion able to become the new Madgod was the Staff of Sheogorath. Sure, he/she could take the throne and have some of the perks, but they were never made immortal. Running out of health still killed you.

If we MUST assume that the player of Oblivion did EVERYTHING quest-wise, they must have died and new Sheogoraths have just been passed the Staff over the 200 years that have passed between then and Skyrim.

...Am I seriously arguing logic within someone else's fictional lore? :{o~

Doesn't matter what race/gender the person that mantled sheo was. Daedric princes change their form at will. Sheo just tends to favor a scottish bearded man.

Also, daedra can still observe mundus from oblivion. They don't have to actually be there. Chimere Graegyn watched the current events from Caecilly in the void. He's even seen things yet to happen by that time.

Sheogorath is the CoC.

One year later, more or less, Sheo returns, saying "Ha ha! Fooled ya! Now, ye didn't really think that you could just walk into a daedric realm and become a daedric lord, just like that? Oh, you did?! That's just wonderful! Even better than I had imagined, and I have a particularly good imagination, if I do say so myself, which I do. Weren't you even the eensy-weensiest bit suspicious when you learned that you didn't really have unlimited daedric power and control over the realm? Ah, mortals! That's why I love you all! Or is that why I love killing you all? No matter ... now get out of my realm. NOW. While I'm still feeling generous. Or you might want to stay and see how long my generosity lasts. That always ends well!"

your pocket protectors are showing

Furthermore, the Sheogorath in Skyrim expresses personal feelings about the events in Oblivion, which is something his predecessor never hinted towards during the SI expansion. It's also possible the prior Sheogorath did actually 'take those eyes when everything was over' as Skyrim's Sheo has remarkably different eyes to his predecessor. They're pure white, rather than black and yellow. However, by far the most important bit of evidence is the fact that no TES game or expansion has ever had their ending rendered 'non-canon'. Arguing against the PC becoming the Madgod is like trying to prove the Nerevarine never killed Almalexia.

Shows how much you know, NERD! My shirt doesn't have pockets!

It has a Pikachu on it.

Sheogorath also talks about a fox and a severed head. That's likely a reference to the Grey Fox and the severed head of Mathieu Bellamont. Both of those are events specific to the CoC
Adding to that, he specifically mentions that he was there for the entire Oblivion Crisis. Infer from that what you will, but he basically makes it obvious who he is.

Also, in response to Uumlau, the old sheogorath doesn't exist anymore. He makes it perfectly clear at the end of the Shivering Isles MQ that he's back to his former self as Jyggalag, and won't ever be the Mad God again