So, we store saved games directly to our site (with special consideration for Steam where there are stored on the cloud), and these saves are referenced by the email address given (which is why we require the email address to save). This is part of the production of the game (i.e., we add all the structure for the sending the saves, provide support of the games, update the system as necessary, etc.).
The problem is that, obviously we can’t let people use our system (that’s for published games), and likewise supporting it for someone else is not something we can do (that would cost us a lot of time and hours for something that we get no benefit from). In practice I think the old *show_password and *restore_password commands are going to be your best bet if you want to host your game for free.
Of course in practice I have to also point out that (assuming your game meets a minimum level of standards), publishing through HG is almost certainly a better choice. Not only do you not have to worry about that, but we’ll provide a lot of the support for the game ourselves, and get to get paid for the game on top of it.