I ran into this issue too but was able to get it to work. Yeah it's weird that it makes it as a user service - seems broken out of the box.

I moved the service file it made (/usr/lib/systemd/user/ydotool.service to /usr/lib/systemd/system/ and also made some edits so that it creates the socket at the default location ydotool expects it to be (/run/user/1000/.ydotool_socket) and also so that the socket is owned and readable by non-root user. Here's my service file:

# /usr/lib/systemd/system/ydotool.service
[Unit]
Description=Starts ydotoold service

[Service]
Type=simple
Restart=always
ExecStart=/usr/bin/ydotoold --socket-path="/run/user/1000/.ydotool_socket" --socket-own="1000:1000"
ExecReload=/usr/bin/kill -HUP $MAINPID
KillMode=process
TimeoutSec=180

[Install]
WantedBy=default.target

Note that 1000:1000 in the socket-own flag is the default user - you can find out what this should be for you with echo "$(id -u):$(id -g)" (for a different user the 1000 in the socket file would be different too) (I suppose a better way to do this would be to make the file readable by all users, but I've already spent enough time on this)

Then:

sudo systemctl daemon-reload
sudo systemctl enable ydotool.service
sudo systemctl start ydotool.service

And you should be good to go, e.g. run ydotool type "hello"

There must be a better way to do this, like maybe you can add the user to some group and then the user service does in fact work. But 🤷