Depends upon the part of the country. In Wisconsin where I grew up, the county government handled a lot of major events in your life, from the draft Board to many different kinds of records and licensing.
As a kid, and growing up in the county seat, the most important thing about a county was the County Fair. Wrote a song about it.. "When you wake up in the morning, you can feel it in the air. It's time again for the County Fair." For Farm kids, the 4-H Clubs were all identified by county. Kids in my county were members of the Rock County 4-H.

Here in Connecticut, I've been good friends for much of my adult life with a couple who lived in the town where I worked. A few years ago, Betsy was hired as Executive Director of the Fairfield County Co-operative Foundation. Her first job was to figure out where the boundaries of Fairfield County are, and it took a little doing. I lived in Fairfield County (and there is no fair, and precious few fields) for over 30 years, and had no idea where it stopped and the next county started. As far as I could ever find out, there was no county "Seat." The only reference to Fairfield COunty I ever saw in the papers was that the cost of living was the highest of any county in the Nation. I would have preferred a Fair, myself.

When we retired, I had one requirement. I wanted to get out of Fairfield County. Funny thing is, I can almost see Fairfield County out of our windows. Cut down a few trees and take down a couple of houses, and I could. I just have to drive about a mile and a half and I'm there. And sure enough, the town right next to ours, separated just by a river, is much more expensive to live in. And we have a farm in our twon, just a short walk away.

Jerry