How do you make the math work with the current empirical data?
An average of 75 mutations were fixed per population in the Lenski experiment. This is less than one per gene over 30 years and is after 50000 generations. The search length you quote is of a single gene/protein of below average residue counts and it still requires 10^8 searches for a single function gene.
If we want to show this is a general mechanism and not specific to bacteria the problem gets much worse with slower generation times and more complex protein functions.