What you say checks out, but it's tight. When I ran these numbers I found that by picking all the right beacons and completing everything, etc, then statistically you should get over the line... but not by that much... which means that a big proportion of people will do the right things but not be lucky - possibly nearly 40% could still fail.I am sorry but thats really on your fellows part.
you can get staffs at a rate of 22 points or lower on average.
with 35 per day * 22 = 770 points
Random pickup 4 logins a day. 20-30 (30 is if you miss 0 pickups) per day = 440-660 points
Quests 4226 points
2x 200 from progress reward.
4226+440+770+200+200= 5836 points
5436/22 = 265 staffs = 13 main prizes or more if you pickup more around your city.
this should be enough for 1 set.
Off course if you constantly pick the wrong "expensive" chests per staff you could screw yourself.
But otherwise all players should be well below 22 points per staff on average.
Do not forget that most of the quest points are in the latter half of the quesline.