Do box culverts handle flow better than culverts (same cross sectional area)?
Well... 1st thing; a box culvert is rectangular. A pipe, is either round or elliptical, although arc pipe exists, we dont use it down here. So, a pipe isnt well suited for a creek, as it either needs to be Huge, or it restricts flow. So; our OP, says 12-24" is normal flow; thats already 50% capacity of a 48" pipe at but its also 8 ft wide; so, we need like 3 48" pipes on a normal day to allow free flowing. If you see a pipe with water up to the mid section in a normal day, or even a monthly rain event, it's too small.
A box culvert, say in the OPs case, a 5ft tallx8 ft wide; gives no restriction on a day to day basis, unrestricted movement for animals, and is only at 40% capacity (based on the 24" flow). I kinda think, based on the range of 12-24"; 12" is normal base line flow; and 24" is a normal rain event? So, day to day, we are at 20% capacity.
Box culverts are Vastly more money than even a 60" pipe. You need to completely dewater the base, add 24" of #57 stone, level it, and crane in the sections. So, equipment alone is much much more. Cast in place... look at any DOT detail for a structure, and you will be dumbfounded by the rebar detail. Even a simple curb inlet... So; even a 32 ft long box culvert; once dry, level, and bedded; figure 2 weeks of rebar work; 2-3 days of form work; false work to support, concrete pump, ect.
Time is money; im guessing a box culvert for his creek is probably double 4x5s, with today's engineering; and around the $150k mark.