I just found this diagram that someone else prepared to try and explain to elected members how transport funding in Christchurch works.

I won’t try to explain it other than to say, I think the main thing it illustrates is how ridiculously complicated it all is. It’s no wonder transport projects take so long to deliver in Aotearoa.
There have occasionally been calls in the past for central government to devolve decision-making for certain types of transport programme to councils: essentially give them an annual transport budget and let them spend it however they wish. For example Wayne Brown in Auckland was elected partly on this policy (here). This would certainly simplify the diagram above, and I think is worth investigating for at least for some of the funding buckets.