OK, hopefully I'm not the only one running into this. It seems that starting a new game in "Full Experience" mode is just completely broken. Everything works up until you select "Work" with Wilfred for the first time. This launches the city map tutorial bit where you have to visit the barracks and tavern. All goes well and runs, but no save made after this point works. Any save made after, regardless of location or status, will always return you to the beginning of this tutorial section when loaded, stripping all progress in missions, story, and equipment. From what I can see it seems that returning players were supposed to be shown this tutorial upon loading an old save, so I'm suspecting there's a variable that's not being set somewhere, maybe something with the save/load script that isn't correctly remembering a variable.
The other game modes appear to work, but it's quite disappointing that the only game mode that allows you to actually do the jobs is unsavable. I managed to get a few hours in and up a few ranks, saving diligently the whole time, only to reload and find that I was back where I started.
I've managed to work around it by changing the default value of the contractFirstTime variable in the LucaniaEvents.rpy file to True. This completely skips the intro conversation with Wilfred telling you to visit the two places. Instead you just go to the usual jobs screen and can use the Events option to enter the city map. Once you do so visiting either of the locations on the map will still play the intro as they would have in the original tutorial, but now you don't have to visit both before leaving the map, and saves made after this actually work. I'm not sure why this variable isn't setting correctly, the code looks like it should be setting, but frankly I'm not familiar enough with Ren'py to be certain.
Hopefully the dev or someone smarter than me can figure it out, but for now if someone else was wanting to restart this great game and play the jobs then here's a temporary solution at least.