Hardcore History is all finished narrative. Where Dan Carlin really shines is taking facts from history, like body counts from ancient battles and making them visceral so that you can really feel how terrified people would have been if there city was being besieged by Mongols, or if you were a soldier in Stalin's army and were used as so much cannon fodder against the nazis.

If you are more interested in the primary sources, he's probably not for you, but if you want to feel history rather than just hear about it, then I highly recommend.

Not a podcast, but I found many of the history courses on The Great Courses series (which is now available on Audible) are like what you desire. Not all of them are, but it's usually possible to tell based on the description.