The need to pay "whatever amount of public money a publisher asks" exists because the important researchers choose to publish in those journals instead of other journals where their articles would be freely available.
If researchers are evaluated on the quality of their research instead of the journal it was published in, then one could simply require that they publish somewhere where it's freely available, but it's a bit of a problem since they need to publish in exactly those journals, which happen to be very expensive.
However, in that case, too, the money is not the researcher's. That said, they are the ones that spend it.
In the case of subscription journals, the argument is similar, except that in this case, it's academic libraries who will have to pay whatever amount of public money a publisher asks of them just to be able to access that research. If the researcher could obtain their credentials elsewhere, they would be able to make their work available somewhere where the libraries and the general public would not have to pay to access it.
It says "If researchers are evaluated on the quality of their research instead of the journal it was published in, that would remove their need to pay whatever amount of public money a publisher asks of them just to obtain career credentials."
But at least in my field, most of high impact-f journals don't charge any money for publishing there. So the researchers don't really use any significant money on the publishing part.
I don't know how exactly they get their profit (other than the obvious subscription fee they charged the school), but I don't think researcher themselves are paying that (unless of course, by "researchers" you mean universities as the organizations, but then again "find ways to promote your academic work that are independent of the journal it’s published in" will not be relevant since you still need to pay to read other people's publications).