The song remains the name

"Bad Company"
"The Monochrome Set"
"Talk Talk"
"Black Sabbath"
"Green Day"
"Icehouse"
"Minor Threat"
"Motörhead"

"Boys Don't Cry"

"The Adverts"
"Wilco (The Song)"

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Is there any example ever of a band covering another band's "Introductory Song?"

Sparks did a cover of "We Are the Clash" for an NME or some other UK magazine's Clash tribute album 10 years ago or so.

Back in the 80s there was an idiotic local punk-funk band in St. Louis that closed their sets with a cover of "Baby Appeal" by the Red Hot Chili Peppers, but since that band thankfully never recorded anything, that doesn't really count.

"Bo Diddley"
"Public Image"
"Kajagoogoo"
"Tin Machine"
"Slowdive"

To a lesser degree:

"New Church"
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It isn't quite the same thing, but there is also a tradition of what I'll refer to here as "Introductory Songs." The song contains the band name, but ups the ante by attaching the song name to the band performing it (in case you needed the dots connected for you):

We Are The Clash - The Clash
We Are The Orange Alabaster Mushroom - The Orange Alabaster Mushroom
Hey Hey We're The Monkees - The Monkees
My Name Is Prince - Prince
Jazz Butcher Theme - The Jazz Butcher

And rather than literally introduce the band, there are also songs wherein the bands simply reference themselves:

The Ballad Of Railroad Jerk - Railroad Jerk
Give It To The Soft Boys - The Soft Boys
The Bubblemen Are Coming - The Bubblemen

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Is there any example ever of a band covering another band's "Introductory Song?" I hope you'll understand why I'm loth to cite The Beatles playing the "Introductory Song" of Sgt. Pepper's Lonely Hearts Club Band as an example.

The closest I can get without cheating is "We're The Replacements" by They Might Be Giants. However, I'm not going to count it, as the tune wasn't written by The Replacements.

"We are not the Village Green Preservation Society" would be good.