The scrollbar is not shown if something switches to the so-called alternate screen of the terminal emulator. It could be the app in question, e.g. if it uses ncurses (which I doubt for something that produces a sequential output, as seen in your screenshot).
tmux also switch to this mode.
I'm not familiar with Minecraft server at all, but a random web find says to run it as
screen /home/minecraft/run.sh. If that's what you do, it's
screen that switches to this mode and not the minecraft
You can run it without
screen, in which case you lose the functionality provided by
screen, e.g. to be able to detact and attach back later. Or you can use the scrolling capabilities (keyboard actions) provided by
screen itself, consult
screen's documentation for what these keys are.