Were the roads of 100 years ago really more strewn with broken glass and lost nails than today?
Nope. However, modern tires have a steel belt in them that stops most stuff from getting through and puncturing the tire.

Back then tires didn't have that, so even a small piece of glass meant you had a flat.
Yep. Plus they were often made with white rubber, which happens to be softer and weaker. It took a little while before someone noticed that black tires lasted longer because the carbon used to color it strengthened the rubber.
Why is Elvish Abe Lincoln wearing a devil costume?

And why is that guy even remotely happy he's doing it, even if his tires are 'puncture proof?'