When I read the OP yesterday the first thought that crossed my mind was be careful what you wish for. Clearly a lot of work has gone into the proposal, and to be fair, it does cover most of the major grievances that have been aired about the existing process. It also has the benefit of removing the angst that the charity nomination process has suffered over recent years.
That said, for me the real essence of giffgaff charity payback is the opportunity for members to nominate something close to their hearts and to campaign on the forum for that charity to get the money. Two such occasions are when Clare's House won, and the last period when Ealing Night Shelters won. On each occasion one member campaigned passionately for their respective chosen charity and the charities got a big boost that they wouldn't have got any other way,
I've looked at Neighbourly and they don't seem to support International Charities (apart form the Netherlands?). I nominated Sightsavers for years who work primarily in Africa and Asia, so they would be excluded, although I never made the shortlist for the vote anyway.
Personally I would rather leave the process as it is with all it's perceived imperfections. It gives members the opportunity to make a real difference to a charity that they feel passionately about.
I don't think many members who have engaged in the charity discussions on the forum will be in favour of the Neighbourly route, whether that's reflective of the wider user base?, who knows.
My last point would be that at the end of the imperfect process we have now, two charities receive a serious amount of money. If we went the Neighbourly route many more charities would receive money, albeit a smaller amount, but in both cases a positive outcome is achieved.