Why can't I play the game corner in Pokemon HeartGold

Unfortunately, it's likely you have a counterfeit cartridge, as that's a common problem for games bought on eBay. This Reddit post has some tips for identifying if it's legitimate or not.

Counterfeit cartridges may seem to play the same as a legitimate copy since they use the real game's code, but because of the different/inferior hardware they often have glitches that a real copy won't (this person for instance had the same experience as you, where the Game Corner didn't work but everything else was fine).

Even in the unlikely case that it's a defective legitimate cartridge, I doubt there would be anything you could do about it since the game is no longer being made. The best you could hope for would probably be to get a refund or exchange from the seller.