Think about it from a numbers perspective. Briefly, $10^{13}$ is a very small number in the combinatorics of protein sequence space.
There are $20^{100} = 1.267651*10^{130}$ possible peptides of 100 amino acids, a rather small protein. That's more proteins than there are atoms in the universe by 50 orders of magnitude. If you like, here's a paper I found via google that happened to pick this same example, they also have some discussion of the sequence space of naturally occurring proteins.
Do you want to do $10^{117}$ such $10^{13}$-plex random peptide experiments to on average 1X sample that 100 amino acid sequence space, or do you want to do one single $10^{13}$-plex mutagenesis experiment on a known protein that kind of does what you want already, which will let you readily sample the local sequence space around that protein?
You can undoubtedly shave a few orders of magnitude off there by encoding your random peptide generative model with non-uniform aa sampling, structural motifs, sequence Markov models, or whatever. But that isn't going to help you very much. You could also argue that you don't have to exhaustively sample the space as long as you get in the locality of something useful. But a useful protein is almost certainly longer than 100 aas, so you still need to sample astronomical numbers of random proteins.
Even if you get lucky with your random polypeptide approach, you are almost certainly only sampling an unoptimized protein, so you will have to turn around and optimize whatever random polypeptide you find.
The existing protein can admittedly probably only get you to a local optimum. But the search for the global optimum is through an incredibly massive domain of possible proteins.
