Do I really have to log a user's IP address and a timestamp after they consent to a cookie?
Log example: 20.05.2018 18:56:32 - 127.0.0.1 - Cookie accepted
I don't even need (I'm not using it in any way) personal data like IP address in my website.
No.
That is spelled out pretty clear in Article 11.
In fact, if you want follow accepted data minimisation principles (and you should), you should not log anything about cookies at your or a third party's website.