Summary:
Can I program a \"thick client\" game in C without reinventing wheels, or should I just bite the bullet and use some libr
Yeah unless you just want to learn all of the details of the things that go into making a game, you definitely want to go with a game engine and just focus on building your game logic rather than the details of graphics, audio, resource management, etc.
Personally I like to recommend the Torque Game Builder (aka Torque 2D) from GarageGames. But you can probably find some free game engines out there that will suit your needs as well.