@Kreig: I think it would be hard to have numbers for Europe as we don't have large national parks like in NZ but a combination of private lands, national parks, regional reserves etc. furthermore the alps for example are split between multiple countries.
One thing I can say is that France is going to have 90 millions visitors this year, for 66 millions inhabitants. I don't know where you found that number for NZ but MBIE says 3.6 millions visitors for last year: http://www.mbie.govt.nz/info-services/sectors-industries/tourism/tourism-research-data/international-travel/international-visitor-arrivals-commentary
If you want to compare to NZ, this is 1.36 visitor for each inhabitant in France, vs 0.77 visitor per inhabitant for NZ, not 1.25 to 1. So yeah it's huge… but not that much compared to Europe.
I can also reply to your question about funding… Huts in France are completely privately funded, either by individuals who own them or by local clubs.
Local clubs can, like in NZ, apply for government and council grants, some do, some don't, to different degrees. So deducing the indirect level of govt investment in club owned huts would be a hard exercise, but having seen budgets of local clubs, the grants they get are usually quite small and don't make up more than a fraction of the budget.
According to https://www.refuges.info/ there are 2383 huts in the European alps (some are free, some are not, some have facilities and food, some are just small cabins, a bit like NZ I guess).
180,000 km of tracks in France are maintained by a non-profit organization called "FF Randonnée". It is mainly relying on volunteers for track maintenance. I couldn't find its budget you have to be a member I think but it seems to be coming mostly from the 3,000 affiliated clubs and their members (about 5 millions € / year if you use the cost of the license and the number of affiliated individuals).
In France national and regional parks are financed partly by the local councils, by the region, by the department and by the government, so it's quite hard to compare budgets with the DOC. For other places that are not protected parks France doesn't have an equivalent of DOC (I would say because we don't have anything left to conserve lol), but they have a national office of forest management (ONF), but it's also managing cutting forests for profit (40% of wood produced in France comes from there) so it's hard to deduce much from their budget as only 91 millions come from the government (out of a budget of 790 millions € in 2011, the rest is coming from the office commercial activites).
So yeah, Europe is very different, as it is mostly relying on individuals through voluntary payment (either in huts or tracks) to finance hiking infrastructure.
I think that it's great that NZ is financing DOC (and huts and tracks, which are great) with tax money, but I also think that it's not enough as it makes it too dependent on politics, leading to a rollercoaster of a budget, uncertainty, and bad management as a result of not knowing how much the budget will be cut the next year… So I'm all for users participating but I find that paying for access to national parks like in Australia is a better idea. As you said $44.45 per person is nothing, and I would much prefer having a simpler system where you only pay an annual/weekly/monthly pass and don't have to pay hut fees, as they represent a lot of added costs in handling the fees (ticket selling and collecting).
Re. reciprocity: anyone residing in France for more than 3 months has free access to free public health insurance, including free dental care. That's far from being the case in NZ ;) (for the record, I find ACC to be a great idea, but it's not replacing a proper public healthcare as it's only for accidents)