I think I know what you're talking about, and Star Realms is a great example.
Playing the AI in SR is like playing the AI in any other game -- once you've done it once, you've done all there is to do.
But there's a pretty neat campaign, too, that serves multiple duties. It's part tutorial, part introduction of new concepts, part story, and part messing around with new cards. It's up to like 14 chapters now, each with 5-6 missions.
Some other attempts at "story mode" off the top of my head would include the Catan series, NH: Puzzle (32-bit, I think), and Warbits. Also, card games often have a campaign/story thing going on -- Elder Scrolls and Eternal are particularly good. The old Magic: TG series had them, too. I'm currently playing the Gwent campaign, although that barely even qualifies as a PC game, since the platforms are limited.