As helene said, 2
♠ should show 4 card support. With only 3 card support and some interest in a moysian, the better way to bid is to do something else economical, hoping to be able to suggest the moysian later.
As for missing the slam, auctions in which both partners hold extras but neither holds a huge hand are amongst the most difficult to bid accurately.
That is one reason for using the 2
♠ bid to show 4: it establishes a solid foundation such that the partnership need never worry that later bids are in search of another trump suit: all such bids can be used to show or deny interest in a spade slam.
So opener's 3
♦ call was correct. He almost has to cuebid over 2
♠ since responder is unlimited. In my preferred style (which I think is fairly mainstream), once we've set trump, the person next to bid must cuebid unless holding an absolutely horrible excuse for a hand. The cuebid doesn't, in itself, announce strong slam interest. If put into words (a useful exercise when thinking about auctions) 3
♦ says: I have a hand at least good enough to cooperate if you like your hand, and I have a diamond control but no club control...my hand may actually be very good but doesn't need to be'
That assumes one bids 1st and 2nd round controls in order of convenience. If you are a '1st round controls then 2nd' cuebidder, the 3
♦call doesn't deny 2nd round club control.
In this scheme, responder has to cuebid because he is in the same position: he has some interest. In addition, he has both clubs and heart controls, and no matter how good partner's hand is, partner will need to know about these controls.
Thus E bids 3
♥.
Things are a little more complex than I have suggested so far. Some people might well play that 3
♦ denies a club control. Personally, I almost never cue shortness as my first cuebid, so I would have bid 3
♦ with a void or stiff club, intending to cue clubs later. I do this because at an early stage I want partner to be encouraged by a K or A in the suit I cue first. The K, in particular, is wasted opposite a shortness cue.
So for me, I don't deny a stiff or void club. These are issues each partnership learns to discuss.
Anyway, if W denied a club control, 3
♥ by East would promise one: there is simply no point cuebidding towards slam if E alredy knows the partnership is off the first two clubs.
Over 3
♥, I would stall with a forcing 3
♠, and this would fetch 4
♣.
One of the 'rules' of this sort of cuebidding is that while the auction is below game, every time a player makes a cuebid, or even the stall of 3
♠, he confirms that what he is hearing and what his hand looks like remain encouraging for slam.
Once East cues the club, West can cue hearts and now East can keycard. I'd keycard not to reach slam: after 4
♥ I would have heard 3 slam tries from partner and I hold 3 Aces and a stiff...I'm bidding slam all day...but there is an outside chance for grand and maybe if partner's diamonds are solid he can bid 7 over a 5N rebid: KQxx x AKQJxx xx is a decent grand and give him the spade J or even the 10 and it becomes even better.
Btw, you can see that in this style, it is often counter-productive to do a lot of jumping over FSF. Indeed, as I have written elsewhere, my view of opener's obligations over FSF is that he should strive to make the cheapest descriptive bid to allow partner to say WHY he used FSF...preservation of bidding space is a bit of an obsession with me
'one of the great markers of the advance of human kindness is the howls you will hear from the Men of God' Johann Hari