What needs to be changed in the save files to navigate through the missions, or whatever is responsible for what, I have already tried everything in different combinations, but I am thrown to the beginning of the game.

I'm gonna be real honest here, navigating through the missions via changing the savefile is both a bad idea and very difficult to do 9because it's a bad idea). Worst case scenario you'll break progression. Best case scenario you'll softlock the game.

The save file tweaks are more meant to unlock specific galleries, add lewd arcana or lewd blooms, unlock and levelup specific items, that kinda stuff. Soft cheat codes, if you prefer.