[clang] [analyzer] Turn NodeBuilderContext into a class (PR #84638)

[Diego A. Estrada Rivera via cfe-commits]

diego-est wrote:

Seems like it wasn't as easy as just making them private. I think I found all the places where there were private member accesses. The tests passed on my side again and the github-pull-requests action. The code_formatter action keeps failing because of the documentation on something I didn't touch (but seems to be fixed high upstream). Everything should be fixed now.

https://github.com/llvm/llvm-project/pull/84638