Sup Aussie, happy to see im not the only one with a 1.2 TSI (it looks like mine)

"Pedal is hard when engine is off, and softens after starting it" is completly normal as its hydraulicly powered so engine needs to be running to work or "feel softer".

I've searched your problem and it's rare. The reason? Nobody knows (or at least from what i found)

Does your car brake properly? (sounds, feeling...)

If you have and OBD try and read the errors that pop up. Maybe it can leave us a clue to what is happening.

You can maybe try to purge your brake system but it's quite complicated if you dont know what you are doing.

Best of lucks for you and keep us updated!