Disclaimer: The OP was posted in "Expert Class Bridge", so I've got to respond there, but nothing I'm about to say should be regarded as expert in any way

... in particular, I'm not about to offer any technical solutons to Jinksy's problem.
Further information: I don't think Jinksy's mentioned that the hand arose at IMPs rather than MPs - in fact, in an English regional (mainly one county) teams-of-8 league match between one of the city's bridge club teams and a team representing its university bridge club. So fuddy-duddies vs juniors ...
I gave the variant auction earlier on because I thought it might provide an initial, simpler way into Jinksy's more difficult problem. It was at our table; I held Jinksy's cards (S) and the auction proceeded with very much the same structure: (P) P (2
♥) X ; (P) 2
♠ (P) 6
♠ All Pass. (I believe that essentially this also happened at the one other table that got to slam; the fourth stopped in game.)
My thoughts on that, at the time and after the event, were:
(1) As I indicated, our X of 2
♥ doesn't give a very precise characterisation of S's hand - it shows 4+
♠s and a hand that wants to get into the auction (thin opening upwards). Having said that, opposite a X that shows 4+
♠ and an opening hand, I think there's a good case, vulnerable at teams, for bidding 4
♠ and not just 3 on N's cards. (The 5th
♠'s a big card; think how much worse the hand could be.) We don't have elaborate continuations available for either N or S (we'd probably muddle or forget them).
(2) I bid 6
♠ as S partly because I felt that if two young E/Ws had nothing more enterprising to say at the vulnerability than an un-raised 2
♥ then partner must have something in reserve. I know we have at least 9 trumps, and I'm prepared to assume that they're likely to play without loss; the
♣s are likely to be worth 4 tricks either through force or by an ruff in partner's hand; I'm pretty sure we're not missing both
♦A and
♦K in addition to the
♥, and partner's playing the hand, so if she has just
♦K x etc it's protected on opening lead; and such additional values as she has must be usefully placed in the red suits. Making the broad assumptions I did about the black suits, I can see 10 tricks in my hand and I'm prepared to play partner for the other 2 without keycard enquiries, which won't help me much with a final decision. Yes, I can find out whether partner has
♠Q (what I can't do is find out if partner has 5
♠s as an alternative to
♠Q) and can enquire either about red Ks or the
♦KQ (but not both), but I'm not sure the extra information to me is worth the extra information to the defenders, so with all the above I judged to bid the small slam and leave it at that.
(3) This route probably gives us the best chance of getting to 7, if partner argues that there's no reason to suppose I'm void in
♦s, so if I can bid 6 then she's got enough extra - in particular,
♦A - to go to the grand. I don't think we're at all likely to get there on a more scientific sequence but, as others have said, getting to slam is the important result and I don't think we should lose too much sleep over stopping at 6.
In your case, the multi opening and the 13-15 balanced option for the double make the auction much less easy, in my opinion, than ours. In this context, I too would bid 3
♠, but not 4, on the N cards. It's worth knowing whether W's pass of 2
♦ X is saying anything specific or merely done in the knowledge that E has another bid, but, for obvious reasons, I have more sympathy than others here with your jump to 6
♠, which seems a very pratical bid despite E/W arguably being perhaps less limited than at our table - it's interesting to see the various suggestions for more extensive methods, but at the table you didn't have them available to you. Similar comments apply to the possibility of N now bidding 7, but on your auction there's more reason to allow for, say, a possible
♦ void and
♥K x in S's hand, so settling for 6 seems natural enough.