Most likely not, but falsification is a nuanced thing. If we have the werewithal to be sure that the falsefication is definitive, we likely are actively working on the replacement already. The perihelion of mercury experiment comes to mind.
In other words, if the randomness pillar were falsified tomorrow, I suspect there would not be a quick agreement on a replacement.
Actually, I'd suggest that this is pretty close to being rather in the same condition as pertained for the perihelion of mercury experiment. I did not originate the idea that mutation could be naturalistically directed. It is an idea that has been crouching around the periphery for a while now because of various timeline discrepencies between the fossil clock and the DNA mutation rate clock, and the untoward apparent longevity of some of the new entries in the Tree of Life.
Your suggestion of random selection from a poplation with a central tendency could be one, but wouldn't there be others?
Well, sure. My personal opinion is that divine intervention or panspermia or both are now, on the available evidence, more likely candidates than spontaneous generation on this planet.
The other question, however, is: "is the candidate ready to be put on the scientific table?" We don't accept default explanations in science. If an hypothesis can't hold its water against the various criteria required of a scientific thesis, we are just going to have the let the problem go until we're way smarter.
IMHO, information theorists (not I.D.ers) will force the issue fairly soon. Driven by the demand to compare genomes - or to treat or prevent viruses, biological WMDs, AIDS and cancer - the information theorists who specialize in molecular biology will be discovering and mapping genetic information content, symbols and algorithms.
It will be quite interesting to see how the biological sciences community recognizes the apparent falsification of randomness in the theory of evolution. My prediction: they will neither declare the theory falsified nor formulate a replacement theory.