I've been looking for a solution for about 45min and I can't find it anywhere (on the internet too). In texture mode, the object has no shading and I can't recognize any shape, which makes it impossible to texture paint.
The same frame in texutre paint mode:
Here's after following the tutorial. At the point, he had shading "on". I did the steps in the video with smart UV project, adding a material that way and so forth: