There are a variety of posts on this site already that address related questions:
I don't believe that any of these answers specifically address the high conservation of the code, except as implicit (though I could be missing something).
A useful paper
However, one such post links to this paper that discusses the evolution of the genetic code.
Part of the introduction restates the question's motivation rather well, I think:
The fundamental question is how these regularities of the standard code came into being, considering that there are more than $10^{84}$ possible alternative code tables if each of the 20 amino acids and the stop signal are to be assigned to at least one codon.
In other words, why has this space of $10^{84}$ codes not been more widely explored than we observe in nature?
The first part of the paper's abstract seems relevant (I have bolded a few sections that might be taken as explanatory hypotheses):
The genetic code is nearly universal, and the arrangement of the codons in the standard codon table is highly non-random. The three main concepts on the origin and evolution of the code are the stereochemical theory, according to which codon assignments are dictated by physico-chemical affinity between amino acids and the cognate codons (anticodons); the coevolution theory, which posits that the code structure coevolved with amino acid biosynthesis pathways; and the error minimization theory under which selection to minimize the adverse effect of point mutations and translation errors was the principal factor of the code’s evolution. These theories are not mutually exclusive and are also compatible with the frozen accident hypothesis, i.e., the notion that the standard code might have no special properties but was fixed simply because all extant life forms share a common ancestor, with subsequent changes to the code, mostly, precluded by the deleterious effect of codon reassignment.
They then review a variety of evidence for each theory, from which I will present one example.
Is the genetic code optimal?
They note a 1991 paper that used biophysical properties of amino acids and estimated that a randomly selected genetic code was ~0.01% likely to be at least as robust as the existing genetic code. In other words, the code seems to have at least somewhat minimized possible errors. So, the existing code is at least well above-average in terms of possible codes, and it's possible that the reason no better codes have been explored is that it's trapped in a local minimum of the code landscape (see Figure 3 from that paper here).
They go on to discuss this in more detail. I personally find other evidence about the "coevolution" and "collective evolution" theories interesting, but obviously it is hard to repeat a multibillion year experiment.