In order to make sense of and manipulate things like genetics we will need to develop machines that can do those things for us. While that's unsatisfying because we generally like the feeling of fully understanding things, such machines will still yield progress and results, which is all we can really hope for here.
I've thought for many years that there are new intellectual tools waiting to be discovered here that will be as big as arithmetic, calculus, or logic. There was a time when humanity had no idea what mathematics -- the whole field -- was, and today there are probably whole analogues to mathematics waiting to be discovered.
Unfortunately we are still in the phase of trying to attack this problem with old ways of thinking. We probably won't even try until we finally come to terms with the fact that the tools we have at our disposal right now do not work to truly understand the genome. This will take a while as humans become emotionally attached to their tools and cling to them. Try debating a programmer on OSes, languages, or editors to see a simple example. :)
Bonus is that once we understand the genome we'll probably understand a lot of other unknown unknowns we didn't even realize we didn't understand. Maybe this is why physics seems stuck. Maybe the cognitive tools we have right now are simply not up to the task of understanding the whole thing.
Edit:
I actually think Stephen Wolfram's doorstop A New Kind of Science was groping in this direction. The book was problematic because of Wolfram's almost comical narcissism (Wolfram sort of tries to take credit for a lot of things he didn't invent), and the techniques it discusses don't seem to have delivered much fruit in and of themselves. Nevertheless at the "meta" level the notion of trying to invent fundamentally new intellectual tools is absolutely what we should be doing. We will of course fail a lot, but that's what happens when you try to do something new.
I'm pretty sure they could learn to write programs. The had algorithms.
We do not have aliens or time travelers to walk us through genomes and fill in the missing pieces of our understanding.
But, if you are suggesting a smart human from 500 B.C. couldn't grasp 'The Art of Programming,' I'd respectfully disagree. Logic and reasoning haven't changed in recorded history. The ancients were no more or less intelligent than the moderns. Whenever I'm tempted to think otherwise, I sit down with my Euclids Elements and see how far into it I can get before I reach the "WTF... How did he figure THAT out!?" An even better cure -- although more recent -- is to see how far you can get through Newton's Principea.
Everything seems obvious and easy in hindsight because we are viewing it with those intellectual tools deeply embedded into our understanding. They are all over our culture and we pick up bits of them as children through osmosis even before we study them formally.
I think getting an ancient Greek or Roman intellectual to understand a large integer factoring algorithm, a proof of work block chain, or an OS kernel would be pretty painful. It would take a lot of tutoring to first teach a lot of things that were not understood in that time at all.
You can sometimes see this today when you see older people in rapidly developing nations trying to learn advanced concepts. They can do it but it takes a while.
My point is that all this assumes a tutor who knows and can explain. For levels of understanding not yet reached by any human, there is no tutor to teach us how to think about the problem.
In short, you are confusing reasoning ability with algorithms designed for a particular form of technology.
Take neural nets. Would someone from the 1970s understand the benefits of a convoluted net vs. a simpler form? Perhaps. But, without the technology to perform millions of training calculations, the lack of comprehension would come from your pupil wondering what the point would be of trying to understand an algorithm that, with his technology, could never be demonstrated, used, or tested.
While cybernetics or the theories of emergence in systems-theory is a step in the right direction, i think the reality of how "something", "works" is way weirder than we expect.
It's much less about how different components superficially "interact" with each other and much more about mind-numbingly complex emergent properties that somehow "happens" between objects, emerge through complex space-time interactions probably with even weirder quantum factors or "fields" that are not apparent by n-dimensional modelling.
When you really begin to dig deep into these problems it becomes an increasingly weird mixture of philosophy, physics and mathematics, revealing endless paradoxes as we increase the resolution.
Basically it becomes like a chicken and egg problem: is the complexity and life-force of a cell or its components in any way able to be "deworlded" or is the cell or other complex organisations simply not able to be understood without the context in which they exists, in other words, the entire universe.
We won't be able to "understand" by looking at the "thing" because no thing is actually at the center, it's not "doing" anything and doesn't even exists outside of our own simple taxonomical concepts. The genome can't be sequenced because it's simply a way to label, divide or reduce a "thing".
Feeding a neural net with enough extremely high resolution data from a space large enough and i could imagine extremely strange patterns emerge. How we would be able to capture such data from reality is the big problem because we simply don't know how many factors are at play. Protein folding is already an incredibly demanding calculation and it's very simple compared to even the tiniest mechanisms of a cell.
Take the genome for example. I really think the notion of gene sequences hobbles our understanding. The genome is not a sequence. It can be sequenced, but that's not what it actually is. A genome is a molecule. Every part of the surface of that molecule is constantly interacting with other molecules. In programming terminology it's like a program where every instruction executes in parallel, always, and in real time. That means that trying to read the genome sequentially like a program or a book is missing the point entirely. We're holding it wrong.
I truly think we are a long way from being able to really "grok" these systems. It took us thousands of years to develop the math, logic, and science to understand complex systems with discrete components and discrete logic. In many ways the digital computer is the apex product of our understanding of discrete systems and discrete logic.
Now we get to figure out concurrent systems and concurrent n^n combinatorial "logic." It might take a lot less than thousands of years because there are more of us and we have a lot more knowledge to work with, but it's not going to be overnight.