Hey, Welcome to the forums…
The player.log file shows that you have multiple corrupted savegames: CHARACTERSLOT_BETA_1 , CHARACTERSLOT_BETA_5.
The way that LE currently works is that if any savegame is corrupted, it prevents the saving of progress in ANY existing game and prevents the saving of a NEW save.
You need to remove the two corrupted files from the Saves folder while Steam & the game are NOT running.
You also need to disable steam cloud saves as the corrupted files are probably in the cloud and being resync’d every time you restart Steam cloud. If you want to use Steam Cloud, then you need to login to steam via a webbrowser (not the client) and remove the problem files from your Remote Storage in Steam cloud itself… If you do not do this, it will just keep syncing the corrupted files back to your machine and continue to cause problems. When you re-enable Steam Cloud after this you MUST tell it to copy local files as the latest… NOT cloud…
As to why the savegame files are corrupted - well, this is a beta and if the game crashed or didnt shutdown properly while it was trying to save a file, it may have caused a problem… Also, the savegames are just plain text files so if you attempted to edit them in any way, even innocently, something could have gone wrong… Sometimes, if Steam itself was interrupted trying to sync the savegames, there is a chance that a savegame could have not uploaded to the cloud correctly.
Once removing the corrupted files as I have suggested, you shouldnt have any further problems like this.
Please be sure to Verify the game files via Steam - just to make sure that your install is ok.