if there is natural glue in the tendon and it can be reconstituted by chewing and saliva, is there a possibility of modifying the modern application process to more closely resemble a "chew and glue" method? utilizing what is already there, and not having to add so much hide glue?
Could be. The thing is that, chemically, hide glue and sinew are ALMOST exactly the same thing, which is collagen. The sinew also contains fibrin. Boiling out hide glue is just denaturing the protein one step back, When it dries it essentially goes through dehydration synthesis, which is not EXACTLY the same process as the living cells use that laid it down, but results in a very similar product. Saliva does the same thing chemically through an enzyme.
The real difference is that we may damage the protein during extraction (over-boiling, etc.) and we lack the precision alignment, matrixing, and perfect placement the living cells are capable of. But, there is glue (collagen) still in the sinew, and when we add more and remove water, hide glue BECOMES part of the sinew, part of what is already there, in a way no other glue does. We just can't do it as well as biology.