Here is a segment that I lifted from Hayashi 2006. While randomizing about 30% of the protein which infected its target was able to get some activity after 20 trials Hayashi estimated 10^77 trials to find the wild type. This is very similar to your discussion.https://doi.org/10.1371/journal.pone.0000096
The question remains regarding how large a population is required to reach the fitness of the wild-type phage. The relative fitness of the wild-type phage, or rather the native D2 domain, is almost equivalent to the global peak of the fitness landscape. By extrapolation, we estimated that adaptive walking requires a library size of 1070 with 35 substitutions to reach comparable fitness. Such a huge search is impractical and implies that evolution of the wild-type phage must have involved not only random substitutions but also other mechanisms, such as homologous recombination. Recombination among neutral or surviving entities may suppress negative mutations and thus escape from mutation-selection-drift balance.