The average height actually went down since the middle ages, and plummeted to below 170cm in the 17th and 18th centuries, making people like Napoleon which was 168-169CM tall about average for the day.
Being over 6" (182cm) in most caucasian / european descendants nations is considered tall even today.
I'm also not sure what he means by finding "historic armor", modern replicas are made to size, even production runs will have a height range that fits the average modern male (or female, there are replica makers that make armor tailored for women).
Whilst being tall he would also have no much problem finding chain mail or plate to fit a 6" tall person, what he would have is a problem finding one in wearable condition which isn't a museum exhibit ;).
Going further back to say the Roman empire we know based on the documents from the period of Gaius Marius (Marian Reforms) that the required height of a roman legionnaire translated to feet was 5"10-6"0 tall.