How can I simplify deploying game server on hosting

how can I simplify this process?

I'm sure you know about Docker containers at this point, and for some reason not yet disclosed you've decided not to go that route. Ok, we can roll with that.

On the CentOS machine run a very simple nanny.sh script:

#! /usr/bin/env bash

# Keeps the game server running, always, even after an unexpected crash.
while true
do
    /bin/game_server  >> /tmp/out.log  2>> /tmp/err.log
    sleep 1
done

Upon startup, have your game server write its PID to /var/run/game_pid.txt.

Write a very simple deploy_new_server_code.sh script:

#! /usr/bin/env bash

set -e
cd some/where/
git pull
make
cp -p game_server /bin/
kill $(cat /var/run/game_pid.txt)

Now from your development client with the IDE, it's just a matter of $ ssh game-server deploy_new_server_code.sh when you push an edit to GitHub.

You don't need to have a screen ssh session running all the time. But if you do, I imagine you'll have a tail -f of the game log running, so you can see how the newly deployed code behaves.