Well for a start is not a religious thing before anyone jumps in there.

None of the major religions prohibit the use of organ donation and their stances are all that it is down to personal choice and preference.
Yes, I expect you would get more. I would wager you'd get the same amount with a co-ordinated health promotion campaign that would most likely cost less than the botched and abortive IT system the government will contract out to keep track of this (with the high likely cockups that will occur and the subsequent law suits).
I pointed out the medical issues to make one thing quite clear - in the lions share of transplantation - organ donation from dead people is not something we should be investing in - we should be investing money in organ donation from living people - why pay money and create all this kerfuffle over a second and possibly third choice treatment for a large proportion of the people needing a donation (obvious exceptions included).
The fact of the matter is not many deaths are suitable for donation - people don't really understand this. Any death that is suitable then the relevant people will be approached anyways.
Lives would be saved by spending the money people are going to waste on this to:
a) Targeted poorly represented communities
b) Facilitating live related programs where possible
c) Increasing the staffing for the dedicated team for families of the deceased.