If I am reading it correctly this would really be “approval with a real consecutive second runoff (not a virtual (basically fake) runoff).” As per:
The question on the ballot explains it all:
Shall the City of St. Louis adopt an ordinance to:
<> establish an open, non-partisan system for elections to the offices of Mayor, Comptroller, President of the Board of Aldermen, and Alderman
<> enable voters to choose all the candidates they wish in the open, non-partisan primary
<> allow the top two candidates to then compete in a runoff during the general election?
I think the wording of it is less than ideal. In all tradition that I know of, “primaries” are private affairs. For example, I and two buddies could always hold a primary, even if we didn’t have an official party. But even if we did have an official party, the primary would not be governed by the same laws as an official “election”. I assume this varies from state to state, but if write-ins are allowed, any non-officially recognized party could hold a primary, choose a candidate in whatever way they agreed to, and potentially get that candidate elected.
So, their open, non-partisan primary is presumably an approval method first round of a true consecutive election. And their runoff during the general election is the single selection method second round of the true consecutive election.
At first glance, it is probably not enfeeblingly oversimple, and definitely not pathologically complex. It would likely disrupt two-party lock-in, but could conceivably be vulnerable to three-party lock-in. It would be much stronger than one-round approval.
I would much prefer the (simple) score method (since approval is inadequately differentiative of choices), but this might, just perhaps, be a good first step.