What exactly is peptone (tryptone?) and why does it not get precipitated?
Sigma-Aldrich has listed long list of their Basic Ingredients of Microbiology, which are called Protein Sources (Peptones):

Peptones and extracts are excellent natural sources of amino acids, peptides, and proteins in growth media. They are most often obtained by enzymatic digestion or acid hydrolysis of natural products such as animal tissues, milk, and plants or microbial cultures. The number of available peptones and extract is enormous, and can promote and sustain the growth of most organisms.

Some protones are animal-based and some are plant-based. For instance, peptone water is a microbial growth medium composed of peptic digest of animal tissue and sodium chloride. The $\mathrm{pH}$ of the medium is $7.2\pm 0.2$ at $\pu{25 ^\circ C}$ and is rich in tryptophan. Peptone water is also a non-selective broth medium which can be used as a primary enrichment medium for the growth of bacteria (Ref.1).
References:

J.E.L. Corry, G.D.W. Curtis, R.M. Baird, Eds, Culture Media for Food Microbiology, Volume 34; 1st Edition, Elsevier Science: London, UK, 1995 (ISBN: 9780444814982).