Finally found a solution to my problem.
In my case I had set a 301 redirect for my front page (that is for “domain.com”) to another page I wanted to use as the front page. Apparently that caused a problem when Elementor tried accessing it’s content (although I’m not sure why it needs the content of my front page if I’m creating a new post).
So I removed the 301 redirect and just changed which page is set as the front page in WordPress.
To clearify: I was avoiding the front page setting because I was afraid what was going to happen to all my backlinks pointing to the page I was redirecting to. But when you change your front page in WordPress, the page’s URL becomes “domain.com” and it’s previous url “domain.com/page” is automatically 301 redirected to “domain.com”. Which means that all links are kept unharmed.
Edit: I finally got rid of the problem. As it turns out, my redirect was only a part of the problem concerning only Elementor. But the ability to save/publish posts in WordPress turned out to be an internal WordPress error. After a few weeks of dealing with this issue I finally got an error when clicking on ‘Publish’ inside WordPress, saying: “The response is not a valid JSON response”.
After browsing around a while I found out that I was far from being alone in this issue. It turns out it is connected to permalinks, I found that this article explains it best.
I was able to fix the issue by either changing the permalink structure to ‘plain’ or by installing the Classic Editor plugin. Since plain permalinks are not acceptable for me, I chose the second. I’m sure this could also be resolved in a more elegant way, but I’m not proficient enough with backend development to do it. Hopefully this will be resolved with future WordPress updates.