IIUC, your argument is assuming the same ribosome mechanism.
Since it is the ribosome which does the binding onto (and so, to some degree, selection of) the base-pair triple, before offering up an amino acid monomer to the growing protein chain, that effectively means that for each triple of base pairs (codons) you need a different ribozyme and a different base-pair sequence. So that would be between 23 and 40-several different ribozymes structures and sequences. Some of the ribozymes could be insensitive to the third base pair in a codon. For example, the codons for glycine, the simplest amino acid, follow this pattern - GGN [Guanine, Guanine, aNything] gives glycine in the output molecule. But whether or not the same base sequence and ribozyme molecule are used to produce those interactions, I don't know. There's no reason to expect it to be so (multiple base pair sequences can produce comparable enzyme activity - which is what ribozymes are).
Theoretically, if there were common structures involved in attaching a new amino acid to a growing protein, then they could use a conserved RNA sequence. But the section that "recognises" the codon are in general different, the section that holds the unattached amino acid residue is of necessity different, the section that forms the enzymatically active pocket (where the attachment condensation reaction takes place is of necessity different ... and that doesn't leave a lot else where there can be much commonality. There might be structural elements that hold the other components in set mutual configurations, but that's equally not a sensitive part of the structure.
But the ribosome, itself, is part of the evolved machinery.
Errr, yes. for 23~40-several ribozymes.
I don't think we have a good idea of just how it evolved.
I think everyone would agree with you there. The evidence has been eaten. Repeatedly. The hints of an earlier 2-base pair coding are suggestive, but by no means everyone accepts that interpretation.
And some experiments have shown that the selection of amino acids used can be altered
Absolutely. The basic structure of these experiments is "select one of the three STOP codons available"; engineer a new ribozyme that attaches to that and adds a "not natural" amino acid into it's place (IIRC, selenium-containing amino acids are popular - they behave similarly to sulphur-containing amino-acids, cross-link protein chains into nets in "interesting" ways, and commit the GMO to living out of the chemistry lab, there being no natural sources of that amino acid) ; insert the biochemical support mechanisms for handling that amino acid in the cell ; and then build some special protein that contains the new amino acid, which would act as a "flag" that your GMO is working as intended. It has been done several times, using different amino acids. You might get your PhD for doing it again, but that would be a journeyman demonstration of competence, not a sign of remarkable brilliance.
So ISTM that live that evovolved independently might will use a different selection of amino acids, and different coding systems.
I think everyone involved with the subject would agree with you on that. The "substitute a 24th amino acid experiments" just described show that is so.
(Nobody seems to be suggesting alternative nucleotides, though.)
Absolutely people have proposed that, and AIUI shown that these can work . It's not just minor substitutions on a purine core (adenine and guanine, and artificial variants) or a pyrimidine core (cytosine, uracil, and thymine) ; other similarly complementary molecule pairs have been designed and tested in the lab and should work. One of the stimuli for such work has been the logical flow "this molecule has been found in space" ; "it has a planar form with reactive bits at the edge" ; "this is similar to DNA/ RNA ; can it work like that?" then showing to some degree of detail that yes, you can build life-like systems using these different molecules.
Life on Earth is composed of a number of polymeric molecules - molecules composed of chains of small, simple molecules. The nucleic acids are one such group of molecules, composed of five distinct "nucleobases" (A, G, C, T, and U ; potentially hundreds of others that aren't used in nature). Proteins are another such group of molecules, made of natural and artificial amino acids (23 natural, potentially thousands of others not found in living systems). Carbohydrates (sugars, more complex carbohydrates, up to cellulose and lignin ; sometimes intimately cross-linked with proteins) are a third group of "biopolymers" with around a dozen basic units frequently used in nature and hundreds (thousands, even ; carbs can get really complex ; they're fully worse than clay minerals - and I say that as a mineralogist ! ) of possible non-natural (or rarely-used-in-nature) alternative monomers. About the only major group of "life molecules" that isn't a polymer of simpler monomer units are the "fatty acids", which lots of organisms use as "energy bank" compounds.
The monomers used by terrestrial life tend to include the simpler ends of the potential ranges of these molecule families (glycine is the simplest amino acid ; glucose is the simplest sugar), but of necessity they extend out into the more complex variants on the particular molecule class, where the number of possible molecules explodes, while the number of molecules actually used remains in the few-dozen range.
Even restricting oneself to a biochemistry broadly similar to ours (nucleic acids for genetics ; proteins for enzymes ; carbohydrates and proteins for structure ; fatty acids for energy storage) there are million, possibly billions of potential biochemistries similar to ours, with very similar overall energy requirements. There doesn't seem to be any particular reason why "our" selection is special.
Then we get onto the complex bit : the structure of the molecules. If you look back on the history of wrong ideas that Crick and Watson (and others) had about the structure of DNA (and RNA, but they didn't really concern themselves with that), you'll see that even within nucleic acids you can potentially have a variety of different structures which could be more stable under different conditions. You evidently know about DNA and RNA. There is also "PNA" (with a protein "chain" instead of a sugar-phosphate "chain") as a possible alternative. Other proposed structures included 4-strand chains (with either doubly-bonding nucleobases, or alternating singly-bonded ones), chains with the nucleobases on the outside of the chain ...
We know well what system is used on Earth. We have no strong reason to believe that that is the only possible system. Even the division of biochemical purposes above (genetic ; enzymatic ; structural ; energy storage) is a distillation of terrestrial biochemistry. But it doesn't really mean that a different way of doing things is impossible - just that we haven't looked elsewhere for an alternative.
If I had my 'druthers (EN_US : "I'd rather that ..." becomes "I'd ruther ..." then " 'druthers" ; same root as "ruthless" and "to rue"), when some spacecraft takes a good sniff of Europa's geyser debris they'll find fragments of a PNA using different nucleobases, and the protein chain made with some amino acids outside the normal terrestrial gamut. Then we'd have good grounds for thinking that life could be "everywhere". (Everywhere with a solvent phase.)