That's a clever system, thanks for sharing it. What I appreciate most about it, though, is the explicit goal of "being easier to count by hand", which I think many advocates of voting-reform overlook.
You even say "by sorting ballots into each of thirteen piles corresponding to the thirteen possible rankings of three candidates", which shows you've actually thought through what the process would look like for the people doing the counting.
When people are proposing voting systems, they really need to consider that their process is being carried out in an adversarial environment, where different groups have incentives to cheat, or accuse the voting system of producing the "wrong" result (e.g. due to the unintuitive outcome, or the complex tactical thinking needed to optimally fill in the ballot), or claim that voting machines were running backdoored firmware (which is a destabilizing claim whether it is in fact true or false).
That last point is perhaps the most important. I think that as time goes on, the pressure to automate the counting process and give instant election results (or worse, do the elections online) is only going to grow, and any voting reform which (if done manually) slows down the counting process, or for any other reason in practice requires people to trust machines that are selected and maintained by officials from one political party, is going to do even more harm to democracy than FPTP does.