It does make sense that Yamamoto can't fully trust Doppel-san. We don't know what exactly she is, after all.
A compulsory club is honestly fine by me. It helps grow bonds between students, and helps them achieve new goals!
It is really cute that Hattori really wanted to be in a club with Yamamoto, to the point that she and Yamamoto were the only ones in class to not be one. Yamamoto's plan of a gaming club's gonna help them bond together. I wonder who the third person's gonna be though.