When I was in college I got one of those credit cards from people giving out free stuff at tables.
It wasn't because I wanted the free stuff, it's because I knew what you have found out: If you don't get establish your credit history early, it's going to be a bitch later.
I got an unsecured card with a limit of $200. I used it a little to order things online (a novelty back then), and kept it for a few years, until I could get some cards with credit limits in the thousands. I used them. Sometimes a little too much, but nothing I couldn't pay off. I don't have any credit card debt right now.
Paid off balances don't count against you as much as you are implying. The single biggest factor is length of revolving credit history in good standing. If you had kept your old card open, you'd have good credit right now.
Yeah it's a flawed system, I'll agree with that. But even 13 years ago, I could easily find out what I needed to know in order to "game" the system. Tell your kids to get those credit cards. They need to start establishing their credit history now; they should have started when they were 18.