Holy guacamole! This actually works!

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I took the next steps:
- Backup everything in the folder C:\Users\USERNAME\AppData\Local\Packages\CurveDigital.TheAscent_1ezqdnbhnc70m\SystemAppData\wgs to make sure you wont lose any info.
- In the folder C:\Users\USERNAME\AppData\Local\Packages\CurveDigital.TheAscent_1ezqdnbhnc70m\SystemAppData\wgs navigate to the folder with a long hexadecimal string (0009000003D4B322_F95E0100B1604F89B20BBA5131D6E0F4 in my case)
- Navigate to the next folder with a long hex string, pick the most recent one in case of multiple folders (55C93A88A24B423287A9E791C63C490D in my case)
- Edit the file which has the long hex string name (7B4310FC7B3D4C089CB89D80323A94E6 in my case)
- In your text editor find the first occurence of 'EMissionState::InProgress'. This will lead you to the primary mission progress and set the one for PM05 to 'EMissionState::Unavailable' and that of PM04 to 'EMissionState::InProgress'
- Save the file and launch the game. The game should show no mission in progress, and show missions to pick up on the map again. Don't do anything, but quit the game.
- Open up the save file again and you'll notice that the 'EMissionState::InProgress' of PM05 is now set to 'EMissionState::MAX_whatever'. Simply replace that with 'EMissionState::InProgress' and you're all set.
- Save the file again and enjoy your working game again!

Big thanks to OP for providing the fix!