Prior to the periodic table, chemistry was a mess, literally and figuratively. The table simplified things by organizing a vast and unruly body of knowledge into a manageable size. The genome has surprized everyone by organizing billions of bits of information into tens of thousands of words. Just as the elements are "islands" of stability, life will be found to involve islands of chemical stability. (This is speculation, of course, but I will stand by it as my prediction.)
Your speculation would be more believable if we were living back before we knew the DNA data content of genes (i.e. long before the genome was mapped).
Data is something that does not depend upon chemical stability, rather, it depends upon the sequential ordering of stable chemicals.
Rearrange the DNA data in a gene and you will change the output form, even though the chemical compounds involved will be just as stable as before.
Ergo, the sequence of the data is more important than the stability of the chemicals.