You're making this much harder than it really is. The results are given to us in reverse order, but the drawing for #1 happened first. Nothing taints the first draw, and the first draw odds cemented each team's chance at winning. Once 15-4 was announced, all we can go on from there is knowing how many combos are left, and how many of those combos belong to the three teams left. It's simple math from there, and everything else is a misunderstanding or an over-complication.
After the 15-4 was announced, each of Buf, Mon, and Car DID in fact have their chances increase at winning #1. Buf went from 185/1000 to 185/310. Mon went from 95/1000 to 95/310. Car went from 30/1000 to 30/310. Those odds didn't change at the drawing though, they only changed in retrospect given how the NHL presented the results. With what we had to work with before the final results, the 59.7% crowd is right, and it's not actually that hard to figure out.
I'd suggest to check my post
http://hfboards.mandatory.com/posts/145312519/ to understand how it works. Let me give you another very simple example too:
Let's say someone flips a normal/fair coin 3 times where each time there is 50% chance of head and 50% chance of tail. The 8 possible outcomes are
(1) 1st: Head, 2nd: Head, 3rd: Head: 12.5%
(2) 1st: Head, 2nd: Head, 3rd: Tail: 12.5%
(3) 1st: Head, 2nd: Tail, 3rd: Head: 12.5%
(4) 1st: Head, 2nd: Tail, 3rd: Tail: 12.5%
(5) 1st: Tail, 2nd: Head, 3rd: Head: 12.5%
(6) 1st: Tail, 2nd: Head, 3rd: Tail: 12.5%
(7) 1st: Tail, 2nd: Tail, 3rd: Head: 12.5%
(8) 1st: Tail, 2nd: Tail, 3rd: Tail: 12.5%
Before we know anything about the result, what's the chance that the first flip was Head? It's 50% and I think everyone agrees with it.
Now, if I tell you the result is all Tails, what's the chance that the first flip
was Head (I emphasized 'was' here since the 3 flips were already done)? Because we now know the result is 3 Tails, the only possibility is (8) so the answer is that it's 0% the first flip
was Head.
Again, before the first flip was made, it's 50%/50% chance of Head. But when we have more information of the result, it's not 50%/50% anymore.
If you feel all 3 being Tails is too extreme, we can use a different example and say the result is 2 Heads and 1 Tail, so what's the chance the first flip
was Head?
When we know the result is 2 Heads and 1 Tail, only (2), (3), and (5) are the possible outcomes. Among these three, two of them, (2) and (3), have the first flip as Head. Thus, once we know the result is 2 Heads and 1 Tail (I want to emphasize again that we know more now), the chance of first flip as Head was 66.6% (2/3).
Edit: before anyone replies that flipping a coin 3 times is different to how NHL lottery works, I want to say that this example is to show percentage could be different once we have more information of the result. For NHL lottery, please see my post linked above.