A few issues come to mind.
one: Economic collapse
Without a strong, global population, you have no strong, global economy. Without a global economy, then things like computers and genomic research quickly grind to a halt. In the middle of your war, resources will have shifted from building and exporting electronics to building and exporting "violence" (See USA's economic shift during WW2: if it didn't help the war effort, there's a good chance it didn't get built.)
So who's building, maintaining, and advancing the massive processing centers required to perform the genomic matching? Who's doing the work to maintain genetics as a field? It takes a massive economy to support the computer industry. And no, existing computers don't last forever; there are parts inside computers that fail over time. Modern servers have a MTBF measured in years. Not centuries. Someone has to keep building new servers, and maintaining the software that runs those severs.
After the collapse, just growing enough food to last through the winter is going to be hard. Especially the first generation or two, as the post-collapse world braves the chaos of no central governments, no grocery stores, no imported goods. I don't know how to reliably feed myself without the aid of commercial food suppliers. Do you?
two: Dark ages
After your war, the population was suddenly dropped to a tiny percentage of the original. How much knowledge was lost during your purge? How many occupations are lost to history, simply because there's no one left to pass on the skills to a new generation? What are the odds that any non-survival skill set will be passed on? At first? sure. But as your war begins to diminish the population, luxury skills like electronics repair or genetics or research in general are going to fade.
Everything gets poured first into warfare, then into not dying. Farming skills and first aid are vital to your remaining population. Medical research, not so much.
By the end, just staying warm through the winter is more important than not burning those books you found in the abandoned university library.
Building things to support your high-tech system requires raw materials we can't simply go pick up. Silicon, rare metals, and complex machines. These things aren't just lying around to be picked up and used. They take a global network of mining operations, refineries, factories to build raw parts, factories to combine those parts into machines... All of that is gone. There's no way a global population can support such long-distance trading.
Immediately after the collapse, when central governments has blown themselves up, there will be a swift and sudden move to self-preservation. People will decide their needs are more important than their neighbors'. Might makes right. After the first few post-collapse winters, no one will immediately trust a stranger. This will make it quite difficult for a city-states to trade, since everyone will be resistant to to the dangers of travel. Sure, your strong, well-armed, nobility might can build their castles and defend them. But building the vast trade networks necessary to maintain a high-tech society is on a whole other scale.