Ok, I think I have figured it out. First, make sure that the folder in your sourcemods folder is tf2classic, not TF2C_2_0_0b1 (It is found within the folder that I just mentioned). Next, to fully restart steam you need to 1. Open Steam 2. Press the Steam button in the very upper left of the application 3. Go to the bottom of the options given in the popup that says Exit 4. Open Steam and it will be in your library.
If you have any questions feel free to ask me.