I see what you're asking here. But you're missing some steps. Allow me to elaborate...
First of all, there two very important qualities of DNA and RNA which make them as significant as they are: they can match up and thereby duplicate themselves, and they can fold into three-dimensional shapes. If they lacked either of those qualities, they would be of little interest at all.
The first property, duplication, can be observed with little interference... even without the interference of a cell. Most chemical reactions proceed because the molecules involved fit together in a certain way that allows their bonds to change and produce different molecules. Nucleic acids tend to force amino acids and other sundry chemicals into a likeness of themselves.
They aren't the only things that do this. Prions are another example. These proteins can produce other proteins like themselves. That is how Mad Cow Disease works - a duplicating protein ends up in your brain and subverts regular proteins which are supposed to be there into duplicates of itself. Prions, however, are pretty much always the same shape.
Not so with nucleic acids. A nucleic acid can fold into shapes. And because chemistry relies on molecules fitting together just so, some of those shapes can work like machines: they can only allow certain kinds of molecules to interact, and they can physically bend and strain them to make otherwise unlikely reactions possible. Chemists call this kind of thing a catalyst - a chemical that makes some reactions go much faster but largely remains the same itself.
So if you start stringing together random bits of RNA and DNA, you will get things that can turn into machines. And some of those machines will do very interesting things, even if most of them are junk. If you find a machine that can then also protect the nucleic acids that made it, or can help those nucleic acids duplicate, or similar things like that, you can bet that over time you're going to see a LOT more of the nucleic acids that produced those machines.
If you think about it, that's all that the massive complexity of life is. Just different ways to produce more of the same DNA. DNA that can support all the complicated functions of a cell has an advantage over DNA that doesn't, so you can expect to see more of it around. So too with cells that can clump together and aid each other, and also with whole creatures the clump together and aid each other.
But don't kid yourself. Intelligence and complexity are OBVIOUSLY not necessarily the optimal solution for making lots of DNA. If it were, we'd see a lot more of it around, wouldn't we? Instead, we see a lot of other kinds of things that are easier to produce and far less likely to cause whole populations to make themselves extinct (like beetles). The jury is still out on us, too.
Just remember - the DNA doesn't make anything massively complex. It just makes RNA, which in turn can make proteins. It is the amounts and interactions of these proteins which produce all the apparent complexity, just as a few keys on a typewriter can produce not only one, but several languages.