Today, we are very certain proteins are polymers of amino acids. We also know that same proteins are bound to cofactors, are linked to sugars (glycoproteins), and have all kinds of other modifications. We know that proteins are made by ribosomes from amino acids linked to tRNA, in a particular sequence determined by mRNA.

There are many analytical methods that provide evidence that proteins are amino acids linked by peptide (amide) bonds. CD and IR spectroscopy shows signals characteristic of the amide functional group, NMR (e.g. 1H-15N HSQC) allows to count the number of different linkages, X-ray crystallography visualizes the complete structure. You can use mass spectrometry to fragment a protein and figure out the sequence. People used to do Edman degradation, residue-by-residue hydrolysis of proteins to figure out their sequence. There are enzymes like trypsin and pepsin that hydrolyze after (or before) a given amino acid, also demonstrating that proteins contain amino acids linked by peptide bonds. 

The OP already traced the very beginnings of figuring this out. It is a philosophical question when the transition was from hypothesizing the presence of peptide linkages to "knowing" that this is how proteins are made from amino acids.