Zero bias means that the grid and cathode are at the same DC potential. If a grid and cathode are close together in DC potential, then you have most of the recipe for the grid to draw current (referred to as grid current). If you have a tightly wound grid (which comes with a high transconductance tube), then the grid current will be greater than a grid with greater spacing. The 6080/6AS7 have a very tightly wound grid, so it's certainly not a great candidate to be zero biased.
When you draw grid current, that will change the bias voltage of the tube, with more grid current creating a grid that's more positive, and therefore causing more plate current to be drawn.
There are metallurgical treatments that can be made to the grid to mitigate this issue, and perhaps that's what's been done to the GYB.