Yes, the dragons and dice are equivalent.
The odds for not hitting a 6 with 6 throws are 33.49%, but only if you have not rolled the dice yet. If you have rolled some of the dice already and not got a 6, then the chance of still not having a 6 after the 6th throw is increased. You should ignore the previous rolls and use the table above with the number of remaining rolls. So if you are doing 6 rolls, and have not got 6 with the first 4 rolls, you have two rolls remaining and the chance of not hitting a 6 is now 69.44%.
If you look at Stratelier’s second graph (above), you’ll see it’s about a one-third chance to have five dragons after 15 eggs, about two-thirds to have all six, and a small chance to have four or fewer.
I like JamesDurning’s approach here. His post includes precise probabilities, and a description of how the numbers are derived. That gives the probability of exactly 5 distinct dragons after 15 eggs as 32.27% and having all 6 as 64.42%.
More than 3% of players still have only four of the dragons after 15 eggs. You have not been lucky, but your luck is not extreme.
Random dragons from eggs is unfun and unfair. I agree that the system needs some reworking (capping at 4 copies, or an ability to craft specific dragons for a larger amount of dragonite).