I don’t understand why you’d stipulate that each would have to be; @glipsnort’s hypothetical case is much better supported by the data than yours.
I’ve been studying the functions of mutant proteins (both designed and natural) for decades, and I don’t know of a single case in which every base or every amino-acid residue must be specific.
For example, we know from the largest data set–hundreds of trials with catalytic antibodies–that one can routinely find a specific enzymatic activity in just 10^8 random sequences (2 x 110-residue variable domains). As these are severely constrained by having to fit in the structure of an antibody, this is surely an underestimate of the typical target space for your purposes.
You appear to be assuming that there can be only one “target.” The evidence says that such an assumption is untenable.