The characteristic impedance of the 1/4λ section is something I was wondering about too, but I question whether 50Ω is where that should be.
Had some stuff typed up here but removed it after questioning myself (common occurrence). I need to go do some math.
Correction: Part of what I deleted treated the matching section like two parts, an inductive lower section and a capacitive upper section. This was in error because the antenna is not connected to the feed point with those components, rather, it is connected at the end of the capacitive section. When the overall matching section is 1/4λ, the inductive and capacitive reactances cancel regardless of where the tap point is meaning that section should not be treated as lumped elements and should be viewed as a transmission line instead. However, I do believe the part where I mentioned how the balun diameter affects bandwidth is still true.