Jagster, don’t take this the wrong way but you are giving bad advice with respect to GDPR on a regular basis.
Wayback Machine / Internet Archive has had some major issues with GDPR and they had to spend a lot of effort to implement the right to delete and to modify.
They also have the status of a “library” since they don’t copy a single specific site. That means that the GDPR gives them “legitimate interest” and they can deal with copyright laws in a bit more relaxed manner.
Yes it would! Your profile picture is definitely PII. By publishing it elsewhere I am taking away your freedom to remove it or change it. It also leaks information about your age and race and maybe even possible physical disabilities. The text that you write is yours, and as long as you don’t remove the post, you have given CDCK a license to publish it (per the ToS). Anyone else publishing it would violate your copyright.
When you create a static copy of a Discourse site, people can come to you and request you to remove their username. That means you would have to go through each file manually or do some fancy search/replace.
If they’re in the EU (EDIT: Or might possibly have users from the EU, or, maybe have users that might move to the EU, or have users that might claim that they live in the EU) they should certainly listen to you. Whether it’s easier to run and maintain discourse indefinitely or run a static site and find out how to do a sed command to do a recursive replacement is up for grabs, I would say.