Well, it's nice to hear from someone trying to defend the logic. Again, this wasn't my experience, so I can't assign motive to anything like that, but let me try to add a little context:
I'm not defending this practice at all, but I think I can understand it. It sounds like the typical problems involved in scheduling 10 people to all perform together at once. Nobody can prevent people from getting sick. Going out of town and missing an interview is pretty flaky, but it happens.
Except it doesn't at other companies. Or if it does, it's corrected before a candidate shows up at the door. In other words, you value a candidate's time. I interview people all the time, and if I have business travel come up, I look at my calendar and make damn sure I don't have any interviews when I'll be out of the office (and if I do, I reschedule. If an emergency happens, I get someone equally qualified to cover). Heck, I even send personal apology emails when I have to reschedule phone screens.
You know why? Because it's polite.
It's certainly human nature to be bitter about someone fucking up an important day for you, but ultimately, bitterness gets you nothing. It's no problem if you want to complain to your friends or on HN, but why not come back for the second try?
Again, I can't ascribe motive to someone else's actions, but it's because it's disrespectful. It's not like, btw, google was apologetic, or said, "You know what, let's do these next round of interviews over video chat", or "We're going to make this up to you in some way". If I recall these events correctly, it was, "Sorry, you didn't get the job because you didn't do well in your interviews". Then a few hours later, "Oh no, wait, you have to do them all again". That's just disrespectful.
Your friend could have had another free trip to Mountain View to actually be interviewed properly, and maybe get the job he wanted.
Who wants a free trip to mountain view? Bear in mind, this isn't a college grad who is just excited to get to go to california, this was someone who was already an experienced product management executive in a leadership position at a middling-size startup. Going out to mountain view messes with their schedule and their work, and again, it's disrespectful.
As for the calculus interview, that sounds like fair game. PMs are supposed to have some PM interviews and some SWE interviews, and math is a perfectly acceptable area to ask SWE candidates.
If the questions had been in the technical area in question, even from a software engineering perspective, I'd be in agreement. But arbitrary knowledge of calculus seems unkind and unnecessary. That's clearly a style preference, though, as you say.
The point I'm generally trying to make isn't about my friend's experience exactly, it's just to add on to what many many people here have said: that google as an organization may treat employees very very well, but treat candidates very poorly.
Coordinating interviews really isn't that hard. Tracking candidate progress isn't that hard. And when you as an employer screw up, picking up the phone and making an apology isn't that hard. Failing at all three shows a fundamental lack of respect for the people you're hiring. You can't just say, "The system is broken".