How to unlock the scene of Katie and the beach, dear author?
If you've unlocked the beach itself, you can invite Monica and Azumi, as well as Agnes and Emily. To unlock the beach, just progress through the main events and the events related to Agnes and Emily. The beach will become available over time.
To be able to invite Monica and Azumi, you need to complete all the gas station events and Azumi’s second event. To access the scenes, you must take over the casino and complete the casino event. However, the scenes also depend on the path you choose with Monica.
As for Katie: to unlock her events, you need to go through the park events and then choose the love path with her. After that, you'll get scenes with her.
The same applies to Agnes: to unlock her new beach events, you need to be on the love path with her and progress through both her side and main quests.