What do other games do?
:: clueless ::
I can only speak for RTS games, but here are some examples:
Supreme Commander, now represented by
Forged Alliance Forever, has a replay vault accessible in the lobby client, which shows replays of all games that have been played, automatically uploaded to their servers. This is dependent on running the lobby client first (this setup is simply a continuation of the 1st-party lobby system that Supreme Commander originally had)
Spring engine games, such as
Zero-K, have pretty much the same system, though instead of being passed through the lobby, it is accessed via web which then triggers the game to open via a lobby (Total Annihilation-based games like their extra-game lobbies). The replays can also be downloaded without having to start the game immediately.
StarCraft 2 doesn't have any auto-upload solution (not a publicly accessible one that I know of, anyway), but tools like
SC2Gears and
SC2ReplayStats stepped in to fill that void. They vary in how automatic they are, and there is a database for each tool, rather than a single centralized database for every StarCraft 2 replay.
For most other games,
GameReplays.org will generally have a section for that game, in which a replay list is available. This requires that the users manually upload their replays, but is the most flexible as it doesn't need any additional tools or dev support, other than enough information about the replay format to parse players, faction selection, game time, and map.