I have been riding off-road since the 1970s, and back then with a standard bike on 26x1.25" wheels the choice was road tires or slightly racier road tires. I had the occasional puncture, but that usually occurred if you rode your tires into the ground or in the vicinity of Blackthorn trees, for example. These days I live in another, sandier country that also has Blackthorns and I would quite happily use something like billy bonkers for summer road and trail riding. In the snow they look like they will clog easily, and if I still lived on clay then they would not be much good anytime it was wet.

The choice really depends on where, when and how you ride.