There’s no alternate way to save - you are right in that when you win a card it should be immediately persisted. The same should happen whenever you transition between different scenes. The Continue button not being available suggests that the save state isn’t being persisted. This is a known issue when trying to play through the browser (Windows and Linux downloads are more reliable) due to the way Unity works with OpenGL. The least supported option is a web browser on a mobile device, but even using a web browser on a laptop/PC has been known to have issues.
I apologise for any frustrations. Unfortunately, this is just a “fan-made” passion project for me. As such, I don’t get much time to commit to it, and tend to cycle back every few months. Feel free to join the Discord server, as I post on there when I make updates to the game