[dontlookdown]

This is cringe-worthy.

But here is what most likely happened:

To get a master (composition) and sync (recording) license to use a song in a commercial, you have to obtain permission from both the writer and the publisher of the song.

In this case, Lindsey is the writer, and he clearly turned down the request.
So the company who made the commercial went to the publisher, and permission for use and (ultimately the license) was granted. But since the writer refused to grant permission, the publisher's only option was to offer the composition ONLY. Their only choice was to get an inexpensive cover recorded or pick a different song for the commercial.

Lindsey obviously sold publishing rights to Go Your Own Way, or he was never a publishing owner - or he was a partial owner and therefore only has partial say in granting permission.

Rights to a song is always owned 50% by a publisher and 50% by the writer(s).
(The performer often has little to no say in permissions and rarely makes much in the way of royalties).

It was very common up until very recently for artists to not own the publishing rights to their songs. The trade off was that it gave publisher's more incentive to get the song out there as much as they could in the way of exposure -- radio, TV, film, etc.

For a very long time Fleetwood Mac refused to grant the rights for any use of any song in film, TV or advertising. But some members switched gears in the late 90's and sold publishing rights to some, but not all of their songs.

I work with a lot of music supervisors, so that's why I'm chiming in.

I wouldn't look at this as a lapse in judgement or taste on Lindsey's part. He most likely has no say on what is licensed for the composition only for Go Your Own Way.

It's a low budget commercial and probably web only (not for broadcast -- or for low budget cable runs), so I doubt much money was made at all from the cover. Maybe a few thousand bucks tops.
The publisher probably made more -- but certainly not enough to put any kid through college.

All of this is worked out through PRO's (Publishing Rights Organizations) - like BMI, ASCAP, etc.