Yes, the board provides a starting point, but it's much more collaborative negotiation - and you (as the incoming exec) are gonna negotiate based on the criteria I mentioned. It's not at all like an IC where you basically get a "take it or leave it, plus a few % more on salary if you push back" offer. It's a different universe.
Is it fair? I dunno. The "intrinsic value" argument doesn't really apply - the CEO is worth $10M/yr if the board thinks they are.
Engineers are too impressed by their own intelligence.
The great irony, of course, is that the messy and immoral world intrudes even there because rational idealists can't find consensus and are subject to the biases that require conventions, institutions, politics and regulation just like the legal and business worlds.
If we were rational you might say we chose "playing with fun toys" over "controlling thousands of people and millions of dollars, and influencing millions of people and billions of dollars". Perhaps, for some people, that is indeed the rational choice if you don't really trust yourself.
CEOs fail all the time, and they aren't shunned but rewarded with golden parachutes and positions on boards of other companies, biding their time until they can "come off the bench".
The management club is something to be admired and I wish engineers would try to adopt some of its behaviors instead of viewing ourselves as mercenaries.