The start of the film is filled with massive plot holes and problems.
The gross violation of the PD is a biggy though. On that day, those people were meant to die in a volcano, by altering those events the crew have chaged fate. They decided to play god and have no clue what their actions will result in. The PD isn't just a suggestion for behaviour, it is Regulation #1 for all Starfleet personnel.
Add to that the need for a starship to be underwater, even though every other cultural study undertaken by a Trek crew has been done safely from orbit.
Then you have NuUhura throwing a strop and arguing with her boyfriend (not to mention superior officer) in the middle of their 'mission', at a time when professionalism would be key.