What is the issue you’re seeing with mission logs? It might not log the result properly if you leave the page before the game over sequence is finished but it seems fine on my browser.
And ya, it could use some quality of life improvements.
Viewing post in The Last Voyage jam comments
It uses localStorage to save, so if it’s aggressive with privacy settings saving in general won’t work (a warning for that is another future item). Does the game reload where you left off if you refresh the page mid game? If it does then something is wrong with Mission Logs. If it doesn’t your privacy or security settings prevent it from saving.