Second, I would like to add that there is something that I haven't really figured out yet... Why did Jobs ask Schmidt to join the board in the first place ??
That is a key question.
Business schools will list the typical reasons for this sort of thing as somehow having to do with every CEO just supports every other, but I don't buy it in this case.
I think you're asking the right question and I'd like to turn to history to see if helps with any speculation.
There are countless posts hereabout on our forums and others about Microsoft and others, but they all seem to miss a few key points.
First - the Apple ][+ computer was a huge market success and it came with an embedded BASIC called AppleSoft.
AppleSoft was was made by Microsoft, and it was the real basis for MS not going under back in the day - and in making the ][+ work, not just be a bare box like the ][ was.
It was a match made in heaven - but Steveo had the upper hand, and Bill needed him badly.
(I lived those days and bought all the early iron, and subscribed to the newsletters back then. What's on the web about the various early computer clubs is all first-hand memory and experience to me on that stuff.)
Then - comes the Lisa and then the Mac. Again with the Mac, Steve is on stage, and he's got three underlings up there - and one of them is Bill.
Spin forward to the 2000s, and what did we see?
After all the fighting and acrimony, what was the first and only browser you could get working under OS X? Microsoft Explorer for the Mac. What was the key question? Could it run Office? (Yep, Office X.)
Later, Safari. Then, out of nowhere - the Google search widget built in to Safari.
I personally think that Jobs wanted to relieve the days of someone with an influential software firm looking up to him, as an underling - and that's why he pulled out a chair for Schmidt.
Probably, I don't know what I'm talking about. But, Jobs was mighty self-centered, and I don't think he ever got over the burn - and I think he thought that he could get lightning to strike twice and people would again praise him the way he wanted.
Oh well - I don't have the answer, but I still think it's a great question!