Incorrect. You can ride on snow without studs, and if you live somewhere where it just snows then melts sure, you can be fine - perhaps Ohio is like that. But if you have ice that sticks around, studs are as much a requirement as 2 working brakes are. Sure, some people have gotten by with 1 sorta-working brake. That doesn't mean it's not a terrible idea.

When I moved back to Minneapolis, MN (about 5 years ago) I sitting around at a stop with a bike club and bemoaning the approach of winter and the end of biking season. Several people piped up to tell me I could keep biking through winter. Having heard this conversation before, I sat back and casually said "Really?...so...have any of you ever had any injuries from winter biking?" (this was when most people have never even heard of studded tires). Every single one of them (at least 5 of them) had broken a major bone winter biking - a collarbone, an arm, both an arm and a collarbone at different times as a direct result of having wiped out on ice...

Studs don't help with snow, but they're the only thing that will give you a grip on ice. No amount of tread or tire pattern will give a pure rubber tire grip on ice. It's also typically dark here if you're commuting in the winter, which doesn't help with spotting ice.

I don't know what the conditions were where you were riding, but studded tires are the one thing that have made winter biking "safe" where I live.