I just unlocked the achievement! Can confirm that they don't all need to be done in one run.
FYI, I unlocked it on my 71st Gold (48 Story/23 Strangers). The 70th Gold was completed in part, in other words, I got a Silver but had completed the other challenges and the 71st was completed as a Gold (I did all challenges at once).
Perhaps just make sure you completely Gold a mission on your 70th one. Not sure if it will unlock if you pass 70 although I have reason to believe it will as is my case in point.
Best of luck!