It's a pity, seems to itch client browser works like 'in private' mode
As for Linux exec export it's easy to implement I think, will try to add it in 0.0.11
The lack of persistent local data for HTML5 games in the itch app is a known issue: https://github.com/itchio/itch/issues/861
It derives from the fact that the game is served from a random http port every time it starts up (for example: http://localhost:23982) - it actually does save data, but the next time you start it, it won't load up the correct data.
The fix for that is probably to switch to a custom protocol (non-http), although that may cause other issues down the line. We're aware of the issue though, so there's hope!