Summary:
Can I program a \"thick client\" game in C without reinventing wheels, or should I just bite the bullet and use some libr
The most common implementation language for A-list games today is C++, and a lot of games embed a scripting language (such as Python or Lua) for game event scripting.
The tools you'd use to write a game have a lot to do with your reasons for writing it, and with your requirements. This is no different from any other programming project, really. If it's a side project, and you're doing it on your own, then only you can assess how much time you have to spend on this and what your performance requirements are.
Generally speaking, today's PCs are fast enough to run 2D platformers written in scripting languages. Using a scripting language will allow you to prototype things faster and you'll have more time to tweak the gameplay. Again, this is no different than with any other project.
If you go with C++, and your reasons don't have to be more elaborate than "because I want to," I would suggest that you look at SDL for rendering and audio support. It will make things a little bit easier.
If you want to learn the underlying technologies (DirectX, or you want to write optimized blitters for some perverse reason) then by all means, use C++.
Having said all that, I would caution you against premature optimization. For a 2D game, you'll probably be better off going with Python and PyGame first. I'd be surprised if those tools will prove to be inadequate on modern PCs.