No they don’t lol. GOG doesn’t even have a client, you have to use Lutris or Heroic Launcher that support it.
Itch has a half implemented Linux client that they gave up years ago and is straight up unusable/broken. The client is worse then a web wrapper and nas no support for Wine, so if the game doesn’t have native Linux support, it just won’t run through the client. It will download exe’s that won’t actually run and silently fail, and doesn’t have any wine support.