I had the same thing, on week one, monday, school, interacted with two NPCs, came at 12:30, My best friend had no interactions even if there was a "Hang Out" tab, and the other two NPCs who you unlock would say they are in a club, but its unavailable, at this point, I was softlocked, and just like you, many buttons where missing, even taking a shower at school just had a button "Return".. this is indeed a sad day.
I am softlocked the same way. I thought the game was forcing me to get the clubs introduced so I tried doing that first thing but now it seems I can only either join the student council (which I do not want to do) or absolutely nothing else. Nothing home, nothing in any of the rooms in schools... just people who won't interact, no option to pass time or even shower...
Edit:
I solved it. You need the tutoring skill (maybe swimming works too?) as that option is still available. Not having the adequate skills will soft lock you.