7 – What will be the consequences of trusting GIT to update Drupal Core?

Background: So I just manually updated ~ 75 Drupal sites (7 and 8) last Friday, when there was another main update. Also, our current development configuration is quite serious. We are starting to implement GIT and we do not have any development server. Yes, that means that the changes in the code are always live …

I am considering using GIT to handle my main updates due to the following article, but I am still quite new with GIT.

My question is how would I implement this on existing sites and is there something that I should be careful to move on?