I suppose there could be a luck factor regarding opening a box containing more than one step, but I wouldn't think the difference would be so significant to jeopardize artifacts. [Or am I missing something?]
You get about 5,000 keys if you complete all of the quests, log in each day, and collect around your city a few times per day.
If you blindly open every single box that's 200 presents without any doubles or the 3&4 present boxes.
200 presents gets you 10 grand prizes of which 1,3,5,7,9 are artifacts. Plus the 2 milestone artifacts for completing quest #20&70 makes 7 artifacts.
With the worst possible luck, all 11 times that you use your 2x bonus results in 11 extra presents. for the other 189 boxes that you open at least some of them will be 2,3, or 4 present boxes so another 9 bonus presents is pretty certain for 1 more artifact (GP #11)
To get that final needed 9th artifact you would need to be clearing boards at least enough to get the bonus presents. (5K keys could clear a dozen boards granting around 40 more bonus presents).
So, if you get the boxes that give bonus presents on each board that you touch you are almost guaranteed a fully evolved owl.
But what about the reshuffle? Well, if you pay to use it just a few times
and never hit a bonus present box with your 2x boost, or miss half of the key collections around your city, or stop after clearing half boards without finding the bonus presents, then you will end up short.
On the other hand, If you think there is an FA coming up that will offer the usual 2 artifacts as rewards, then you can hit reshuffle up to 33x almost doubling your daily prize take and still get 7 artifacts guaranteed plus those 2 from the FA for a fully evolved owl.