I rented a car and bought insurance, but even if I had zero-deductible,
it would not apply at all to the "undercarriage".

What is the undercarriage? The chassis and all the suspension parts?
The exhaust sysem? Dents from below in the floor panels.

What sort of damage can one do to the undercarriage? Other than
busting a ball joint, they seem pretty reliable.

What does one have to do to do that damage to the undercarriage?

And how hard do rental companies check whether there is undercarriage
damage? If it's not creaking when you drive in and not crawling
sideways, are you okay, or do they actually look underneath?