Is there a way to get more "gametime" on a specific game in Origin after it runs out? Or a way to extend one's "gametime" before running out?
No, you CANNOT extend your OriginTM Game TimeTM counters.
Game time is designed to be a timed demo. It's basically, "try before you buy". Extending is not possible due to the way the system works and how timed data is stored.
There isn't any method to extend it because it is a timed demo of a full game.
Alternatively, you can do what the system's trying to achieve, which is to buy the game.
With this system, they're hoping that you're invested into the game by the end of the two hours, to the point where you'll want to continue progress, thus buy the game.
Couldn't they just do a time extender?
That'd the pointless.. It is a timed demo. Not some subscription-based model placed on a game that can normally be purchased for a one-time fee.
I'd dread to see what happens if Minecraft had adopted a subscription-based model.