There are two other ideas to throw in here.
1) just to add to KAM's thoughtful answer. There was also a thought that the last base also gives a lot of flexibility for GC content which responds to some
2) lets not forget that redundancy in the genetic code helps give some resistance to mutations which might be disruptive. the amino acids less disruptive to a typical protein fold are more common in the code. (we have some idea of this from studying mutations of protein structures.)
3) some biochemists have proposed that there is a sense that the 20 amino acids we have are a fairly stable set - that adding other amino acids don't help create better proteins. Peter Schultz learned some of this as his group really wanted to add extra, human synthesized amino acids into native proteins.
I was at a talk where he noted that attempts to make cysteine with a longer side chain caused the amino acid to cyclize to form a thioolactone.
Thinking along these lines adding another CH2 group to proline might not make packing better. There is probably some value, but just not enough to disrupt all the sensitive machinery for making and realizing the genetic code.