I'm looking for a quite simple bitcoin-based donation service that would keep track of the $USD sent to a particular address/wallet and display that on a private url page.
Does anyone know if such a site exists?
As said by above posters there is no such site. But there is a easy way to set this up, just create a address for only the purpose of donation, then any bitcoin sent to that address would be visible on the blockchain and you can keep track of it via any of the block explorers.