First done by University of Michigan of course. 
You can think that if you want, but the preponderance of evidence disagrees. My version of the story is a hallmark of baseball lore and has been written about time and time again. That’s not to say that legend is always right, but baseball is as well reported and chronicled as anything and I’d be surprised if this story could have the legs it does if it’s not largely true. Baseball geeks love nothing more than proving each other wrong.
The song dates back to 1908 and has always been well known and popular at the ballpark, but the tradition as it is now is different than it was before Harry and Veeck. In any case, the Cubs originated the PA music concept as well (prior to 1941 there were barbershop quartets and roving brass groups but no stadium wide music) and it’s almost certain Take Me Out… was a feature. I have no doubt it was sung before Harry, but it was a more laissez faire affair.
The honor of dotting the “i” is reserved for a senior sousaphone player each year. In Ohio, if you are a tuba player, that’s the highest aspiration one can have. IIRC, only two people have ever dotted the “i” who were not a sousaphone player. One of them was Jack Nicklaus, a graduate of The Ohio State University.
I’d vote for the Green Jacket, for what it is worth.
The preponderance of evidence consists of Cub fans’ and Chicago-based writers’ opinions, who researched the topic by asking people at the Cubby Bear Lounge. SABR hasn’t weighed in on this topic to my knowledge. If I’m wrong, cite me to the info. I say Cubs/Wrigley Take Me Out to the Ballgame announcer sing-along tradition is limited to the Cubs and Wrigley. The other clubs play the song during the stretch, and the fans sing badly without the benefit of an announcer leading them, but sometimes having the words printed on the scoreboard to sing along to, as they’ve always done it.
Wikipedia lists some others, but you’re right that it’s a very rare honor indeed. Other notable guest dotters have been Bob Hope, Buster Douglas, and former OSU coach Woody Hayes.
Regrettable, but true. 
One tradition that did begin at Comiskey (in 1977, I believe) was “Na na, hey hey goodbye” sung by the crowd led by Nancy Faust on organ. Steam, the band that had a #1 with it in 1969, presented her with a gold record in gratitude. That team also popularized curtain calls after home runs. Hal McRae, in ridicule, waved his helmet all the way around the bases after a home run for the visiting Royals one day in July. That second game of a doubleheader averted a four-game sweep for the Sox, and cut their lead to four-and-a-half over the Royals, who would eventually bury them. Still, the tradition did catch on some around the league.
I can’t tell you how hard I laughed at this.
What in the love of god are your arguing? First you say everyone has always done it, and now you’re saying the announcer sing along is a Cubs thing. Which is it? The Cubs started the tradition of having a pipe organ. The Sox and Harry Caray started the tradition of leading the crowd in song.
Is this really in doubt? Or are you just pissing into the wind because you can?
Me too. I was living in Liverpool when Shankly died and when they were winning their European Cups.
I got a similar feeling when the Welsh rugby fans sang their national anthem - and that was at Twickenham - it must be quite something else in Cardiff, as shown here.
Rene Rancourt singing national anthems in Boston.
I think this is common among the Big Ten teams as well. My Michigan State band has always done it, even if the other team’s band is there. In that case each band will play the other’s fight song which is kind of fun since they play it just a little differently. The MSU band leader said on a radio show that the band really gets a kick out of it when they do it for a bowl game, the other school’s fans aren’t expecting it and react very positively when they hear their school’s song.
I was in Northwestern’s marching band. This is correct in that all Big Ten teams play the opposing team’s fight song, both in pre-game shows and post-game shows. And this is done even if the opposing team’s band has travelled to the game. And often, the bands will do a joint post game show, in which frequently each will play the opposite’s fight song as part of the post game.
Though, everyone has a different arrangement.
Strawberries and cream at Wimbledon
Women wearing utterly outlandish hats at Royal Ascot
Three Lions sung at England v Whoever football matches
That reminds me;
This year the Montreal Symphony Orchestra celebrated its 75 years, and they put on a show at the Bell Centre which intertwined symphony music with the 100th season of the Montreal Canadiens. I didn’t go, but I’ve seen footage, and I wish I had been there, because taking a symphony and mixing in sports/Habs traditions resulted in things like;
Na na, hey hey goodbye
Ode to Joy in CAC jerseys (poor video, sorry, mais cé beau en ostie!)
The Hockey Night in Canada theme song by an orchestra
The Wave (skip to about 3:40, because the filmer’s voice drowns out everything else anyways… in fact, turn off the sound!)
and the Olé Olé Olé to the Habs veterans (sure, we know it started with soccer, but Montreal kidnapped it and isn’t giving it back)
I think it’s cool how so many sports traditions become such a part of a culture, like the baseball ones in the States and hockey ones here (with some overlap, of course).
How about the crowd going crazy during the national anthem at Hawks games? Do any other teams’ fans do that?
My personal fave is probably Chief Illiniwek doing his dance at halftimes. What do you say…?
That WAS my favorite. 