"Also, doesn't "meat tenderizer" work, essentially, by speeding up the decomposition of meat?"

Decomposition or rot or putrefaction are terms usually applied to the process of decay caused by bacterial or fungal action.

Meat tenderizers are enzymes that break down proteins, including collagen, a connective tissue which accounts for much of the toughness in meat. They do that by cleaving peptide bonds between amino acids found in complex proteins.