Looks like these days the best solution for controlling an Android phone from a PC is scrcpy. It is being actively maintained, has very high performance (framerate, latency), is available cross-platform. It also does not require root on the phone.
To do it myself I would sure put what you need for rsync or scp in a shell file and put the command in cron:
Shell : crontab -e
Write : */5 * * * * /home/bly/get_newest_stuff.sh
You didn't mention what you want to display... videos, images, text, run loader for a running presentation program.
At least you can open images with ImageMagick.
# display whatever....