I think the main issue you’re having is that in your Style Guide, you didn’t style a Rich Text block (or if you did it got deleted but the classes were modified).
Right now, within rich text elements, all links are set as display: block. I believe if you change the styling to display: inline, that this should solve your issue. Because they are display: block, everything is stacking on top of each other.
Just to add, you may need to create two different Style classes of Rich Text elements if you’re wanting some links to stack vertically and other to stay inline (i.e. article headings vs inline links). Then you just apply the appropriate class to that rich text CMS item.