No, there’s no doubt in my mind that the median age of a president is greater than the median age of the general population, but that in no way implies that old age “runs deep” among presidents, which I take to mean that it’s a common thing.

About half are less than 55 and half are more than 55. Most are somewhere near 55 (since the distribution conforms to a bell curve). 55, in my opinion, is not “old age”, but your mileage may vary.

Extra context: a 55-year-old who gets elected for president twice would finish at 63, about 3-4 years short of retirement age (66-67 depending on birth year).

The population of the U.S. has grown considerably since 1985. The median age of the entire population does not seem like a good metric. What's the median age of people qualified to run for president today? It appears to be 50-60.

As the minimum age to become president is 35, you should exclude people under 35 from the statistics. I can believe that 55 is close to the median age of people 35 and older.