> the original biocatalysts a.k.a. enzymes were RNA molecules unlike anything that exists today.
Why “unlike anything that exists today”? Catalytically active RNA is still ubiquitous and well preserved across kingdoms (they’re some of the most ancient mechanisms for which we have phylogenetic evidence).
> it is far more likely that the first enzymes were non-ribosomal peptides
Peptides are good catalysts but they lack a high-fidelity copying mechanism so they’re not good raw material for evolution.
Taken together, your hypothesis seems to be based on a disbelief that RNAs lost their catalytic capability (which isn’t actually the case), but conversely you readily accept (without any evidence) that peptides used to have replication capabilities that were lost without a trace.
> Non-ribosomal peptides exist even today in all living beings and they are assembled in the correct sequence from amino-acids without depending directly on the information stored in nucleic acids, like the proteins.
Can you give an example of a catalytically active polypeptide which is assembled without any RNA template? I can’t think of any.
> Memories, like nucleic acids, are certainly not necessary for designing an ensemble of molecules able of self-replication
No, but what’s necessary is some kind of structure that records bits of information in some kind of order, and a mechanism for reading and writing this. Individual oligopeptides are fundamentally insufficient, you need structures capable of larger assemblies, and the information of these assemblies needs to be recorded. RNA elegantly solves both these requirements. DNA solves the storage requirement but is a very bad catalyst. Peptides solve the catalytic part of the requirement but are bad information store (prions can store specific configurations but they don’t seem to generalise).
It’s hard to overstate how important this information storage requirement is for evolution — arguably much more important than even marginally efficient catalysis. To the extent that some (admittedly far-fetched) hypotheses for early life even posit carriers such as clay minerals [1] which have, to a close approximation, zero catalytic capability — simply because they’re such an attractive medium for storing information (in the form of crystal lattice surfaces), high fidelity replication (through deposition of a new layer of minerals), and mutation (through structural modifications of the surface which are carried over through generations).