Is there any method out there for reheating coffee that doesn't make it taste burnt or old??
The problem is not only the reheating, also the storing.
Maybe quick freezing, with liquid nitrogen or so, immediately after brewing it will help.
I knew someone who would brew a pot of extremely dark coffee every morning, and then cool it immediately. She made cups of coffee from that syrup all day long. Adding hot water and for those who liked it, hot milk. Her coffee was famous. She did not only cool it quickly, also kept it as airtight as could reasonably be, without too much hassle. (Old method, from at least begin last century)