The general term for for the fused glass-like material formed during a nuclear test" seems to be "atomsite" - https://www.osti.gov/pages/servlets/purl/3001767 . Some images of atomsite from the Trinity and Semipalatinsk test sites are at http://www.radioaktivitaet.uni-bremen.de/downloads/Pittauero...
As a wild guess, what's special is it might be the easiest to get, with many samples to study.
As I recall, research on meteorite impacts use the similarly formed Impactite to deduce various things about ancient impact sites. As an aside, I think they also do really elaborate calculations of force and angle of impact based on surveying the spread pattern and distribution of these little glass chunks.
and when a quick search finds the relevant research paper at https://www.sciencedirect.com/science/article/abs/pii/S22133...
"A complex association of millimeter-sized, aerodynamically-shaped debris, including glass spherules, glass filaments, and composite-fused melt particles was recovered from beach sands on the shores of the Motoujina Peninsula in Hiroshima Bay, Japan. .... This study interprets the large volumes of fallout debris generated under extreme temperature conditions as products of the Hiroshima August 6th, 1945 atomic bomb aerial detonation. The chemical composition of the melt debris provides clues to their origin, particularly with regard to city building materials. This study is the first published record and description of fallout resulting from the destruction of an urban environment by atomic bombing."
and when the dynamics are described in https://www.sciencedirect.com/science/article/pii/S0012821X2...
"Our analyses support the hypothesis that the Hiroshima glasses are gas phase condensates formed in the nuclear fireball ... The Hiroshima bomb exploded 580 m above the city so that no crater was formed at the surface. The plasma (called fireball) formed at this altitude, had a maximal radius of 230 m (Imanaka,2011; Malik, 1985). ... The maximum temperature on the ground during the explosion was estimated to be 6287°C (thermal wave) (Radvanec, 2009) while the maximum pressure reached 35 tons per m² (3171 bar) at the arrival of the blast wave at the hypocenter, 1.3s after the explosion (Radvanec, 2009). Under such conditions, the city materials were injected as vapor or melted debris in the air 0.5 to 2 seconds after the explosion and vaporized by the high temperatures prevailing in the plasma (4000-2500 K) at that time (Adams et al., 1960, Supplementary S6)."
I think what your analysis didn't consider was the observation that "The fact that no crater was produced by the explosion (Glasstone & Dolan, 1977) reinforce the idea that the primary source of Hiroshima glasses is the city materials and not the soils or the basement rocks below the city."
I have some Trinitite. It's easy to obtain. You can buy it in rock shops, especially in the Southwest. Back in the day, when they opened the Trinity site to the public a couple of days a year, there was no prohibition on collecting Trinitite. That's why rock shops have it.
They still open the site to the public one or two days a year but they won't let you collect Trinitite any more. You can walk on it but don't try to put it in your pocket or a person with a gun will pull you aside for a chat.
https://www.nps.gov/thingstodo/alamogordo-visit-the-trinity-...
There's not much public access to the sites in Nevada; at least until the DOE resumes their bus tours and they're even more hard over about not letting you pick up anything.
Obviously it's historically significant, and the new forms of matter were first discovered there , so that's why trinitite is named after the site. But 80 years later, wouldn't we expect the other bomb sites to have just as many interesting chemical reactions?