What has caused life to choose this unfathomably tiny subset of all possible proteins?
The answer is chance or, even better, contingency.

About your calculations, it is true that the theoretical sequences are almost unlimited, but the basic scaffolds are not. Very different sequences can fold into the same basic scaffold and have a similar reactivity/function. So, even if not all the sequences have been explored on this planet, most of the scaffolds and functionalities have been. It is true that protein-based life could be evolved somewhere else, and it is true that the overall sequence space evolved somewhere else would be different from the one evolved on this planet. However, I would expect to find almost the same palette of biochemical reactions that we have discovered on our proteins.

I would suggest reading the work of P.L. Luisi here. He makes really good points on this issue.

A new paper just came out, I think it is a nice addendum to this answer.