Why is a monitor less likely to damage a computer? If the monitor
proves bad, as indicated by the open on a pin that should be grounded,
won't I still have to connect a good monitor to the neighbors' computer?


It's not, it's more likely as it contains much higher voltages right on the
same logic board as the low voltage stuff, but the chance of damage is so
remote it's hardly worth mentioning, you're more likely to damage one or the
other by dropping them.

What might be wrong with a computer's video card that would damage a
monitor? I could check the video outputs with a scope. Would that
assure me that it was safe to plug in my monitor?


Nothing, not on a reasonably modern monitor anyway. Some of the early ones
would fry if fed an invalid sync signal, anything made in the last decade at
least will just shut down.

I thought I'd connect a good monitor and see if the computer worked.
What's the point of disconnecting everything in the computer without
first knowing that anything is wrong?

There isn't one, though if it doesn't boot you should disconnect anything
non essential and see if that allows it to boot.