Yeah a systemd unit is the obvious solution on systemd-driven systems indeed. It is not what the original install.sh does, so just take care to not double things.
rc.local on modern Debian/DietPi is btw as well started via systemd unit 
: systemctl status rc-local
But it requires the rc.local file to be executable, which is not the case by default.
Okay, so we have the script started now but it does not lead to any OLED output 
.
Does i2cdetect btw show expected output, so that I2C is really enabled?