Why don't proteins form branched polymers?
Note that branching is not unheard of, for instance glutathione is an abundant short branched peptide formed by condensation of the carboxylic group on a glutamic acid side chain to the main chain amino group of a cysteine aminoacid. This requires a dedicated enzyme both for synthesis (glutamate cysteine ligase) and degradation of the gamma peptide bond. Such gamma peptide bonds could serve as the basis for branching but afaik occur only in special cases.
An important example of branching also occurs during ubiquitylation, in which ubiquitin is attached to a protein's C-terminus via lysine side chains to target the protein for degradation. A variety of polyubiquitin chain arrangements (including multiple branching) have been identified.
I suppose one can only speculate from its absence (or non-ubiquity) that branching either (1) does not generally confer a significant evolutionary advantage or that (2) there are evolutionary barriers that have impeded development of advantageous biochemical machinery for generation and regulation of branching.