It is functionally identical, so long as the neutral isn't broken. They
connect to the same terminal strip at the breaker box, after all. If
the neutral is broken, of course, all bets are off, and this is why I
brought it up in the first place.

Doesn't matter. The rules are the the ground and neutral are bonded at the
service entrance or the breaker box. Not afterward.

It's no longer code but most older homes with an electric dryer or an
electric stove used a three wire plug. The neutral was connected to the
chassis of the dryer or stove. In new installations the chassis is
connected to a separate ground wire. In theory the "new way" is safer, in
practice it doesn't make any difference but electricians don't have a
choice.

The basic idea behind the grounding scheme is that the protective ground
only carries fault currents.