The effect of "punishing" is not linear, it only matters if it reaches a threshold.

No. Even a little amount of punishing can weaken relationships and reduces the amount of information that gets transferred. It can also produce time lags.

Fear of punishment is also not the only reason why a politician might want to prevent knowledge of a local epidemic in a remote village from spreading. There can also be inner political reasons.

If you look at the original article by the CDC person, they speak about lack of trust that might make it hard to distribute vaccines. While trust is a resource that doesn't really grow linearly it's not about threshold effects.