I've worked at a phone repair shop for the last 2 years, and I mainly work on iPhones. Very, very rarely are home buttons broken outside of a repair. Usually when they are broken the phone isn't in.... great condition. In fact I'm not sure I recall a home button ever being broken without the rest of the phone being completely shattered.
Having the accessibility mode isn't uncommon, I only see phones for a short amount of time and mostly do a quick test, so I'll admit if a button is intermittent I might not catch it.
I will say breaking home buttons during repairs aren't uncommon though, it really depends on how honest the repair shop is. Even veterans can have issues with the home buttons because their cables are _sooooo_ thin, they're glued in place, and even the tiniest tear can mean it's done for. Touch ID can never be replaced and as of 7's and up they cannot be functionally replaced.
That's a reasonable point, but it seems they've chosen to illustrate the point with an example that isn't true.
That was the first time I heard of the iPhone home button breaking.
Because they break easily. The home button is probably one of the most commonly broken parts of an iPhone, so the X doesn't come with one: ta-da, can't break it any more. Plus, if it's on a screen, you can have an unlimited combination of buttons whenever you want - think of the screen real estate freed up by using an on-screen keyboard on the iPhone compared to a Blackberry.
For what it's worth, I agree with you that buttons are better. But their disappearance isn't a random occurrence.