You literally can't finish everything currently is in the game without doing the mall rescue event. Some of the content is locked behind that.
What you need for the event:
- Play Ochaco's route until she can be recruited. (You can bring her with you into the cave lab)
- Play Mei's route until you finish the cave lab. You should've got the hardening Quirk by now.
- Finish Tsuyu's route in either way (blackmail or not)
- Play Jiro's route until after watching the movie together with her (not required, but recommended)
- Then go to sleep at night, you will get a prompt asking if you want to process with the main story. ---> This is the mall rescue event.
If you still can't have it unlocked, upload a save file onto our discord channel and I'll take a look at it.