"One benefit of such reactors is the relatively short life of its waste products, which would be in the hundreds of years as opposed to millions of years for existing nuclear reactors."

This line in the wikipedia is outright false. There's no way for a fissioning nucleus to magically know if the neutron that hit it came from an accelerator or another fissioning U atom. Neutron's can't be spray painted green to greenwash / eco brand them to magically make them better than non-greenwashed neutrons.

The rest of the article is more or less true. It misses the two most important uses of accelerator driven reactors. The first is even if you can't generate primary energy and its a net energy sink, you can still transmute elements. So if nasa needs more of specific Pu isotopes for a RTG, you can incredibly slowly create it by burning an enormous quantity of coal without having an actual reactor. Or you could make a plant that overall eats (lots of) electricity and squirts out americium for smoke detectors. The second use is obviously research, you now have a magic machine that eats electricity and while it eats electricity it creates an environment very much like the inside of a real reactor, without the costs and size and weight and security concerns of a normal reactor. Not exactly the same but probably good enough for short term materials science or other fooling around.

Um, I have no idea about how this particular tech works, but neutrons with different momenta can cause dramatically different transitions in the nuclei they strike. No magic required. Is it clear that these neutrons are supposed to have the same momentum as those produced in traditional reactors? That would be surprising.

Yes fasts and thermals generally have varying cross sections. The graph of fission products don't vary much. The "well known" ability to identify which reactor a used rod came from, is because the fission products themselves react to neutron flux, and for materials science reasons no one runs rods to more than a couple percent burnup and long enough to stabilize.

So a rod isn't just 1% used up (a vacuum?) and 99% pure and unused, its first hour of fission at a certain rate resulted in a certain random distribution of nuclei which themselves were irradiated for precisely 2345 hours of full power operation, and then hour two's fission products got irradiated for 2344 hours, repeat. Its a massive summation. What makes it even more complicated is some of the product nuclei are themselves radioactive with a pretty short half life (this is where the ten or so percent of decay heat energy comes from ...) so if you shut it down from hours 1023 to hour 1025 then the end result product will be quite a bit different because you're missing two hours of product isotopes, missing two hours of irradiation for the old isotopes, and the old isotopes decayed naturally for two hours (perhaps into different isotopes, some of which might be radioactive).

If you're really bored, and want to run a big numerical simulation, and have programmatic access to a nuclei table (or just make one up with random data?) you can have all kinds of fun simulating a reactor and summing, decaying, and reacting hundreds to thousands of isotopes under varying conditions and then seeing what you get.

Maybe a TLDR is there are a lot of ways to properly ignite charcoal for a charcoal grill, but the end result cooked steak tastes about the same in the end.