I've never tried to take a rental truly off-road but I've definitely had them on very unpaved roads they probably shouldn't have been on. You can get true off-road vehicles in locales like Death Valley but they're very pricey compared to a standard rental.
I've definitely had a number of minor scrapes etc. over decades and I've never once had an issue--though I've almost exclusively dealt with the larger companies for whom minor scrapes etc. are presumably considered normal.
Nitpick: unless I just popped into an alternate timeline, Kauai is American [1].
[1] https://en.wikipedia.org/wiki/Kauai