Board Game Arena

I'm premium and really happy to support the site, but really that's the only reason I'm premium (there's not a huge benefit for actually being premium that I can see)

One feature that's lacking and actually tempting me to other sites, is not having access to replays. I love being able to review my games afterwards and save the really good ones, but I can't and I'm getting increasingly dissatisfied with not being able to

This feature is already available, and for all members (not just premium).

From your account page, scroll to the "games played" section low on the page (or click the link under your avatar).
Clicking on any of the games will bring up the summary screen for that table.
On each of them is a "replay game" link that will replay the game.

All games can be replayed by anyone with an account, and all you need to do to save a particular game is save the table number.