I think I never thought as much about it as while writing this reply so thanks for asking. I also think I didn't even use the term "war dramas" before but it's very accurate to describe what kind of shows I like to watch lol.

I think shows that try to be historically accurate and portray characters as (flawed) humans interest me for three key reasons:

Thanks to the historical aspect, you can basically see the technological progress in action. We went from fighting with sticks, bows and horses to bayonets, rifles, tanks, artillery, bombers etc. The maximized incentives (life or death scenario) are a catalyst for such progress. Michael Saylor in the Saylor series on the What Is Money show by Robert Breedlove goes in depth in this technological progress. One phrase that Saylor said in this series stuck with me: "There is no such thing as a fair fight [in nature]."²

But war fortunately not only accelerates the ways we can kill each other but also other technology that is useful in many ways. For example, cryptography and cryptanalysis got pretty important in WWII due to the widespread use of radio communication which can easily be intercepted. This was naturally very interesting to me. Some sources say that breaking the Enigma code was crucial for the Allies to win the war. Stuff like this isn't usually mentioned in shows but the (counter)intelligence part is. For example, mission briefings are all about intelligence: What kind of information do we have? How was this information gathered? Is it trustworthy? Is it accurate? What do we do with it?

You can also use shows about wars to get generally interested about the era it happened in since wars are like the climax of politics.

For example, I believe my "real interest" in wars started with playing Rising Storm 2 which is a first-person shooter set in the Vietnam war with an emphasis on realistic mechanics (one shot usually kills). It had a "Woodstock experience" setting where your visuals changed to being on Acid. This made me look up stuff about the Summer of Love, the War on Drugs, the Nixon Shock and the end of the Bretton Woods system which all happened in the same political era. So you could say I learned about the history of money simply by playing a video game and looking stuff up related to it.

Similar things are true for shows like Band of Brothers and The Pacific.

Shows only give you a glimpse of it since screen time is limited, it depends on the actors and there is no good way to go into detail what a specific character is thinking without breaking the flow of the show. But as with every good show, character development is still important.

I found that reading books from veterans is very good to have empathy with them. Again, thanks to RS2, I read my first such book. It was Chickenhawk from Robert Mason who flew Hueys in a transport role during the Vietnam war.³ It was a great book and it got me hooked on reading more from other veterans. I especially liked it because it starts with his relatable desire to fly helicopters and the US Army gave him this chance since they would pay for his training.⁴ However, while training he did not imagine that he would fly helicopters while getting shot at one day. The vietnam war broke out after he graduated where he then developed PTSD and had to give up flying because of that. A very tragic story including the treatment of such veterans after the war (which relates back to the Summer of Love and the politics of the war).

What is also interesting is to watch a show and then read the book on which the show is based on and spot differences. Sometimes, these differences are only in detail, sometimes, they are pretty big. The show was already pretty brutal but then you read the book and you're like... oh, that wasn't even close, lol.

I also like to watch interviews like this one from Major Dick Winters.