Do mandatory installs require the full download amount? I thought it was like some Switch games where some is on the cart and the rest is downloaded.
The mandatory installs copy the entire content of the disc to the HDD. Afterwards the disc's only purpose is the certification process that you have the game that you installed to the HDD. The console plays all games from the HDD, so the disc is basically just a way to copy a game to your HDD without an internet connection.
If the disc didn't need to be inserted to play the game, one person could buy the disc and give it to an unlimited number of other people.