you expect to wait 2x the average.

Actually, this is also slightly incorrect, conflating "expected wait time until next block" with "average current block duration". These are two different expectancies: You always expect to wait exactly the average wait time of 10min until next block; and you expect your block to be of double average duration (20min, with the instant of sampling lying somewhere in the middle).

The concept of memorylessness is not very intuitive, and thus from the true statements "average block time 10min" together with "the current instant is expected to be somewhere in the middle between two blocks" is drawn the wrong conclusion (disregarding memorylessness): "average wait time is 10/2 = 5min". So, new attempt at a tl;dr:

"you expect to wait 2x the intuitively (wrongly) expected time (of 5min)"