Parts of the RNA world theory are correct, but other parts are completely bogus and completely illogical.

What is correct is that RNA must have existed a very long time before DNA, during which RNA was the only nucleic acid.

Moreover, self-replicating RNA must have existed before ribosomes and proteins (where "protein" means a polypeptide that is synthesized using a RNA template).

It should be obvious that neither ribosomes nor protein-encoding RNA-sequences may exist before the existence of self-replicating RNA, because the living being in which those would exist would immediately die without descendants, together with its content of ribosomes and proteins.

So far so good, but some of the supporters of the RNA World theory claim that before the existence of protein-based enzymes, all chemical reactions inside a living being must have been catalyzed by RNA molecules.

This is an illogical claim, which is false beyond any reasonable doubt. Some RNA-based catalysts may have existed quite early, and some still exist today. However, any RNA-based catalyst could have appeared only at a later time after the establishment of RNA self-replication. The argument is the same as for protein-based catalysts, any living being with a RNA catalyst, but without RNA replication would die and the RNA catalyst would disappear without descendants.

So there is no doubt that the first feature of RNA that has appeared was self-replication, and at that time RNA could not have any other role inside a living being, because any such role would not have been inherited.

In other words, the first self-replicating RNA molecules were a kind of RNA virus, which multiplied inside the existing living beings, consuming energy and substances, without providing benefits. Only later, when eventually RNA templates have become the main method for synthesizing the useful components of a living being, something akin to a symbiosis between RNA and the rest of the living being was achieved, arriving to the structure of life that is known today.

For the first self-replicating RNA molecule to appear, the living beings must have contained abundant ATP and the other nucleotides. So the original role of the nucleobases in living beings was not the storage of information, but the storage of the energy required for synthesizing organic polymers. The self-polimerization of the nucleotides, which forms RNA, was an unwanted side reaction. In other words, before the RNA world, there already was an ATP world, which was the first user of nucleobases.

If RNA could not have been the material for making enzymes before the proteins, such enzymes must have been made from peptides (i.e. polymers of amino-acids), exactly like the enzymes of today, but those peptides must have not been synthesized using ribosomes, like the proteins. Such peptides still exist today and they remain widespread in all living beings, and they are named non-ribosomal peptides. Their mechanisms of synthesis are much less understood than the mechanisms of RNA-based protein synthesis. It is likely that more research into non-ribosomal peptides might provide a better understanding of how a living being without RNA could function.

In order to have a self-replicating living being you do not need a self-replicating molecule able to store arbitrary information, like RNA. It is enough to have a chain of synthesis reactions that closes a positive-feedback cycle, i.e. the products of one reaction are reactants for the next reaction and the products of the last reaction are the reactants for the first. If the chain of reactions produces all the components of a living being, growth and self-replication can be achieved.

The defect of such a living being is that evolution is extremely difficult. any mutation in one of the catalysts used in the chain of reactions is more likely to break the positive feedback and lead to death, instead of producing an improved living being. After the appearance of memory molecules, i.e. RNA and later DNA, which can store the recipe for making an arbitrary polymer molecule, it became possible to explore by mutations a much greater space of solutions, leading to a greatly accelerated evolution of the living beings.