"Do 2-inch 12s actually throw the wonderful, flat, and even patterns as they are reported to do? Does anyone know where this might have been documented?"

Shot is shot and once it's out of the bore it's all really the same and bore/length has in reality nothing to do with any patterning or perceived time of shot arriving on target.
I have had a 1949 H&H 2" in hand but the asking price was simply way too high.
There are Purdey's around but I haven't seen one or better still had one in my hands, yet.
My 1941 A&F 2" box lock (Tolley gun at 5# 13oz) is listed in the same A&F period catalog I have with 2" Purdey's offered by A&F.
A joy to carry, recoil is sharp with GameBore 2" loads.
I much prefer a 5# 9oz 16 gauge Fox that CSMC built for me as it's to my dimensions, single trigger but that has nothing to do with the gauge/bore.
The 2" Arietta's are simply very heavy and do not have any of the attributes of a true Brit 2".
FWIW they are NOT making any more Brit 2" box lock guns, maybe H&H or Purdey would make you one if you came up with $$$$$. Let me repeat in case anyone reads any drivel on any website, they are not making any more 2" Brit box locks.
So if you just want a 2" Brit gun and you find a decent example, purchase it but it will not do anything a 16 will do and the other way around.
I haven't read the 16 ga Forum for years and I like 16 gauge guns!-Dick