This is one of the things people have trouble accepting, but it does work.I suspect the same thing happens here, but never to any degree that has actually caused a problem with blockage or flow.
We get long winter stretches of 15 - 20F overnight lows with 30 - 35F daytime highs. So, average is below freezing, and ground stays relatively frozen, other than the top 1/2" taking the sun which softens each afternoon. The daytime sun will melt everything most days, with temperature anywhere near 32F, so water does flow into the pipes and probably freeze when it hits the cold pipe surface a foot or two underground.
We also get a few stretches most years of 0F to 20F days, but they rarely last much more than a week at a time, and then we're back up to that more typical 20F - 30F range.
Obviously not ideal, but it has just never caused a problem, at our latitude. The OP is in Ohio, so probably very similar weather to us, if not a little warmer.
I am in Ohio, and where I live, there are no storm sewers.
Every home in my area discharges their downspouts into the drainage ditch at the street, which is less than 2' deep. My downspout drain's run about 15"-20" deep at bottom. You couldn't put them in below the frost line, on most properties here, if you wanted to.
Just like my sanitary drain, which is 12" deep, water isn't stored in there, it flows through it. Flowing water doesn't freeze, or, occur in down spouts during very cold weather. And, if the discharge is installed properly, the flowing water actually melts snow, and ice near it.
Yes, there is the potential for issues, if the installer doesn't pitch the pipes, or fails to allow for the discharge point to have sufficient area as to avoid the water pooling there. Like, if the pipe discharges in the bottom of a poorly draining ditch, that has standing water in it. Even then, the weather has to provide the right conditions for it to be a problem.