If everything is clickable, nothing is clickable – how to improve text links?

I am working on an app that shows quite a bit of textual information. None of it can be safely removed without losing the meaning (icons that you see improve scanning of the page, but still require details that are provided in the text).

Every bit of text is a hyperlink, and (almost) every link takes the user to a different place.

Here is how it looks:

When a user mouse-overs a link, it gets underlined:

The same happens for the links on the right side.

The problem is that people are not clicking virtually any links. Some of them didn't even consider trying the text on the left.

Chevrons on the right side (external links on the screenshots) attract attention, and some users click on them.

I have tried making all links underlined by default, but then the page looks ugly - literally everything gets underlined. I have tried adding chevrons after all links - it leads to the same issue where everything has a chevron. Removing chevrons completely makes the page look like a report.

And this is the general feeling that I got: users perceive the page to be a report, and don't expect that its elements are clickable. Reading about affordance didn't help me at all.

My question is this: when all text elements on the page are clickable, how do you make text them obviously clickable without making the text unreadable?

Update: on the mobile devices layout changes to this:

The three-dot menu visible here is cut-off on previous screenshots.