`nix flake update` with private flake inputs in github action

I have a flake with a git+ssh://git@github.com/... input which points to a private repository. When accessing this from a github action, I am rewriting the ssh access to https via git config: git config --global url."https://git@github.com".insteadOf ssh://git@github.com which works great for nix flake check. now I want to bump my flake inputs in the CI with nix flake update but its not working:

remote: Repository not found.
fatal: repository 'https://github.com/myPrivateRepo/' not found

Why can the nix flake update command not find the private repo via https when nix flake check can? Is the git config ssh to https hack not working for nix flake update? The error message is showing the https url so the rewrite appears to have worked. Is nix flake update missing the access permissions to the repo although nix flake check has access?

I think nix flake check git clones all repositories to ~/.cache/nix/gitv3 and I assume that nix flake update does not invoke git when updating the inputs. Could that be the reason why the git config hack does not work?

I know that the action works when I am specifying the flake input as https. However, in the dev teams that I have been in so far we always used ssh to access our private repos. Does anyone have a different experience?

Do you have access tokens set for github? nix.conf - Nix Reference Manual