Unfortunately, I can’t give a very specific answer to your question, since I don’t use Genki or Bunpro 
I do like the structure of textbooks, but I used NEJ and am currently working my way through みんなの日本語. So my experience with the reading sections in those textbooks may be completely different from yours, since I don’t know how Genki is structured.
However, I don’t think it’s necessarily a matter of gaining “more” from them as it is to gain a “different” benefit. In the textbooks I used, the reading sections are to reinforce the grammar point and vocabulary you just learned, and so are structured in a way that repeats that grammar point in various ways to reinforce your learning. This mostly applies to NEJ, since the みんなの日本語 text I’m using doesn’t give little stories, mostly dialogue exchanges to read over. And I’d say one of NEJ’s key flaws for me is that it’s geared towards university students, so the reading sections are catered to that, whereas I don’t get much value out of talking about different departments and dorms and classmates and studying abroad, etc. because it doesn’t apply to my life stage. But they did their job of reinforcing the grammar points.
However, graded readers are books. Very simple picture books that are intended to tell a story in the most accessible way possible to someone who may have only just learned hiragana. This is, unfortunately, what makes them “boring”. In order to be interesting reading material, they would have to flood the reader with too many new words and new grammar forms, which would render them inaccessible. It’s different from reading example sentences, because you’re actually reading a cohesive story, just simplified, rather than a standalone sentence.
They are short though. Most are less than 10 pages. So they’re not a huge committment if you do try one out.
You don’t have to read them at all if you don’t want to. I found them to be a convenient, free stepping stone to stuff I actually wanted to read.
When I was just easing into reading, my preferred method was to read a graded reader on Tadoku and an NHK News Easy article. The graded readers I used for more passive learning, to train my mind to recognize the cadence of Japanese sentences and how they’re structured, note the usage of particles in the pared down, simple sentences, etc. The NHK News Articles I analyzed, trying to figure out the meaning by picking out what few words and kanji I knew before I looked anything up. Predictably, I understood practically nothing. (Link to me and Pandoravakarian debating whether the article was about bear populations or bear attacks).
I like to read a lot of things from various sources for different purposes.
But that’s just my method. Yours may differ!
Another source of easy (free!) material to supplement your reading is Watanoc, which you may or may not find more interesting. It doesn’t look like it’s still updated, but they have a backlog of news-style articles about food and cultural events that you can sort by level (N5, N4, etc.)
Well, I mean, it is in a different language that we don’t know very well yet when we’re starting with graded readers. And yeah, I was absolutely intimidated and frustrated when I read the first few level 0 graded readers and understood maybe half of it, if we’re being generous.
And yeah, you definitely have to look up stuff when you’re first learning to read. A lot of stuff! All the time! Which is time consuming, frustrating, and takes a lot of the enjoyment out of it. It’s a necessary evil.
But my point for the Tadoku graded readers in particular is that they recommend for you not to look stuff up while reading because they’re designed so that you don’t necessarily have to, since they (generally) aim to get you to understand the unknown words through repetition and context. And their explicit purpose is to get you to read as much as possible in as short a time as possible, because the more you read, the more you absorb, even if you don’t fully understand it. And if there’s still stuff you don’t understand by the end, you can always go back and look it up, or write down your unknown words as you go and look up any you still don’t know by the end, or any method you please. Whatever you have to do to keep going and not get discouraged and stop.
For me, I like to read, so stopping all the time to look stuff up is a huge motivation killer. But at the same time, it is necessary. So I compromise where I usually have one really easy thing to read where I’m okay with just getting the gist, and usually 2 things (usually book club reads) where I have to actively look up vocabulary and grammar forms in order to have the slightest idea what’s going on.
(I wish I were as motivated for listening, but alas, I enjoy books and comics and hate podcasts 
)