If contributors pledge less than their marginal benefit (which they would), these contributors win whether the DAC raises enough $ or not. Contributors therefore do not care whether or not the good is built, and therefore, whether or not they are pivotal. They can't lose.
You're missing the case where users' contributions could be superfluous.
No, it is impossible to make a superfluous contribution. Even donations of one cent are incentive compatible. (By "contributors", I was referring to cash-donators ["contributors pledge"]).
It is wise of you to ask for help, you do seem to be very confused. Let's try an example of a simple AC first:
 There are 10 people.
 9 of the 10 each value a Lighthouse at $5. They would be indifferent between getting $5 cash today, or having a lighthouse magically appear today. Choosing between $4 and a Lighthouse, they'd take the Lighthouse, and choosing between $6 and the Lighthouse, they'd take the $6.
 The 10th person values the Lighthouse at 0 (they don't want it).
 A Lighthouse would cost $20 to build today.
Clearly, no one will privately build the lighthouse, because each individual will calculate ($5 > $20) = FALSE.
It is also clear that, under an AC
, each of the 9 individuals would contribute $1 at t=1 (as this increases their utility by +4 * Probability(SuccessfulFundraise), which is always a positive number). At t=2 they might increase their contribution to 2$ each, then $3 each. 9*3=27 would be raised, and (as 27 > 20) some would proportionally be refunded (7/9 = $0.78 to each of the 9 donors) and the lighthouse constructed. Each of the 9 donors would have his or her name inscribed on the inside of the lighthouse. Each would benefit (5-(3-0.78))= +2.78, in other words, as purely a result of the AC existing, each individual would gain happiness equal to "the happiness they would have gained from magically receiving $2.78 right now".
Note that nowhere did anyone calculate their probability of being pivotal, nor anyone invoke the CLT (which would be inappropriate as ((N=10)<30). Even if we had a total of 100 people, 90 valuing the L at $5, and 10 at $0, the players could not themselves evoke the CLT, as they cannot observe any preferences other than their own
. They would have N=1 observation of P (which would be 1.00 if they were in the group of 90, and 0.00 if they were in the group of 10).
Suppose that they could (!) magically -and unrealistically- observe some kind anonymous distribution of P. They could average 100 P's and get a single N'=1 observation of a normally distributed y. Y would have a mean of .9 and a sd of .09 by CLT, but what would anyone do with this information? They could look directly at the distribution of P, and learn much more.
For the Dominant AC
 For liquidity purposes, all individuals everywhere do not enjoy it when their money is tied up. They dislike even making a pledge (which they would get back if not enough money is raised) which locks up money for a single day. They dislike being in this state of affairs (that of a locked dollar) for a single day as much as they dislike "permanently losing $0.05 during the course of a single day".
Now we have a problem, because the 9 can no longer increase their utility with certainty. In fact, possibly none will contribute.
This is what the DAC solves. One new 11th individual says: "I will risk my own $10, to try and raise $32 total ($12 for me, $20 for the lighthouse). Our contract runs all day today and tomorrow." The loss is bounded at $0.10 / dollar, yet gains from the entrepreneur's 10$ could total (5/31.999)*10 = $1.56 for a donation of the contributor's full $5 which "almost made it but didn't". The contributors are back in win-win territory, all 9 donate $3.56, $32 is raised, and the entrepreneur gains $1.50 = $12 - ($10 + (10*.05*1)) (the contract ended during the first day) and the 9 contributors gain $1.44 = $5.00 - 3.56.