Some of the amino acids found in proteins encoded in the genetic code are also found in experiments that simulate the prebiotic Earth as well as in chondritic meteorites. Some suggest that chemical selection acted on this type of molecules in early stages of the evolution of life, and then the biological selection acted and gave rise to amino acids such as histidine and tryptophan. The reality is that the genetic code and the selection of certain amino acids over others is still an unsolved question.

I recommend these articles:

https://link.springer.com/article/10.1007%2Fs11084-013-9344-3

https://pubs.rsc.org/en/Content/ArticleLanding/2016/CP/C6CP00793G

And of course the seminal work of Crick and Orgel:

https://www.sciencedirect.com/science/article/pii/0022283668903926

https://www.sciencedirect.com/science/article/pii/0022283668903938

Why are only L Amino Acids physiological? Because evolution selected them through 'natural selection'!

I am not an expert at proteins biosynthesis but I have some experience with phylogenetics and genetic engineering. I am a bit curious about your task here because somehow this question feels odd.

If you look at known pathways for synthesizing and degrading amino acids, others than the canonical 20 appear. L,L-Diaminpimelate and L-pipecolate for lysine, L-ornithine of arginine, L-arogenate for phenylalanine pathway, etc. Proline biosynthesis passes thru an unstable alpha amino acid, L-glutamate semialdehyde, which would be a quality that would probably make it unlikely to enter into protein synthesis.

Signaling pathways in plants and non-ribosomal peptides in bacteria/fungi provide a lot more. You can find all sorts of variations there on the canonical 20n- side chains a bit longer, beta-lysine, etc. plus D isomers, sometimes epimerized after being incorporated into peptides.

Once the genetic code was set, adding another amino acid would require recoding of an existing codon. If that codon was in use, then such a substitution could be harmful. So perhaps we see the first 20 to grab codons, with the addition translational amino acids requiring a different challenging evolutionary route, adding in context to translation, in order to make it. Codons do occasionally become unassigned in organisms with extreme codon bias due to extreme GC bias, so if you were to go looking for new deviations on the canonical code that incorporated new amino acids, those might be good hunting ground. But it is easier to reassign a codon to an amino acid already being used in translation (aminoacyl transferase already exists), so such events are likely to be exceedingly rare.