I must say that I am thrilled by your approach! Though I do agree with the others that "the real Germany" is hard to define. I assume the wording came up in contrast to what we Germans feel is a dis(ney)torted view on Germany, with sugar frosted castles and Lederhosen.
I wouldn't want to name a region of Germany as "the real Germany" because then we'd basically be falling into the same trap as the Bavaria/Neuschwanstein fans, only for another region. So I'll only very cautiously name one city that - IMO - might represent "the real Germany" better than most others: Berlin. You'll find WW II and pre-war and post-war history there, you'll find the traces of the wall and of East and West Germany, you'll find ancient history in the greatest museums, you'll find an extremely lively and exciting young Germany. So if anything, yes, I think I'd go with Berlin.
Now let's have a look at your interests and those sites you named. You will find pre-1600 sites and monastic history everywhere, so that in itself isn't anything to base a decision on. You will find a pre-1600 church worth looking at in the most remote and unimportant village. :-)
Let's try to do a bit of pre-sorting. For a first overview, I'll try to sort your favorites very roughly by region (you might argue about some of those placements - let's not be picky for now):
- North: Hanseatic League, Bremen, Lübeck, Stralsund, Wismar
- West: Cologne, remains of the Roman Empire
- West/ Central: Corvey
- East/ Central: Quedlinburg
- Southwest: Ice Caves, Black Forest, Maulbronn, Messel, remains of the Roman Empire
- Southeast: -
Now isn't that fascinating! You didn't name a single site in the Southeast. So wherever the "real Germany" might be, it's not Bavaria for you!
If I were you, I would make a basic decision either for the North or the South. Since you say your preference is seeing one geographical/cultural area at a time, I assume that you would not be happy trying to rush through both.
So, for the North, that might include a Hanseatic focus on a roughly circular route:
- Bremen
- Lübeck
- Wismar
- Stralsund
- Berlin
- Quedlinburg
- Corvey
- If you have spare time left: Cologne
You can easily take out some (like Bremen, which is a bit out of the way), or add others that are practically on that route, like the castle and half-timbered houses in Wernigerode, or more UNESCO world heritage churches in Hildesheim.
Instead of the Black Forest, you would have some other great natural attractions pretty much on that route or very close by:
- Baltic Sea
- Rügen
- Mecklenburg Lakes region / Müritz National Park
- Harz (great for hiking!)
The Southwest alternative would include (in no particular order for now):
- Ice Caves
- Black Forest
- Maulbronn
- Messel
- Remains of the Roman Empire
- You could easily add to that a trip down the Rhine to Cologne.
Now obviously, these are not cut-and-dried route suggestions, but they might help you find a starting point that is right for you.