First let us be clear on our terminologies: the amount of BIAS CURRENT (using that term literally) is indeed actually very near to zero. The act of "biasing" a tube places a negative (relative to the cathode) "choke off" or "control" voltage on the first grids of the output tubes. Without that "choking" action, the large positive voltage on the plates and the near-ground or near-zero volts on the cathodes would result in those output tubes passing MASSIVELY more current than they can safely handle. The plates would glow red and in short order, if the fuse did not blow, those tubes would burn up or otherwise destroy themselves in any one of several ways. When this condition exists or partially exists, we call this "redplating".
So what I am saying is that if you are of an engineering background you may be in fact properly understanding the term "bias current" but in large part, we do not care what "bias current" actually is. It is miniscule. Bias only places a charge on the control grids and within the bias circuit itself, there is (or there should be) literally "no" current.
But what *is* absolutely salient is that this charge, this amount of volts, acts as "gatekeeper" on the current flow from cathode to plate. We measure the CATHODE or PLATE current in the tube which RESULTS FROM the application of that gatekeeper bias voltage. And THAT we care deeply about.
We generally assume that ALL current, every single electron passing from ground, up to the cathode and then transiting the tube, landing on the plate, is a unity. In fact, some current is bled off via the screen grid connection, but for 95% of our activities, we tend to ignore that current and simply assume that cathode current is the same as plate current. That's all for now on this small point.
So I hope that distinction becomes clear. We frankly do not care about "bias current". We adjust "bias voltage" to produce a known range of "plate current" in the whole tube.
So now how to measure this plate current? We generally do not like to measure current, because it forces breaking a circuit somewhere. And traditionally, the way you destroy meters is to measure current. We like even less the idea of interrupting our plate connection because big volts reside there. The cathodes are lashed to ground in a stock Fender, so we can't measure (without interrupting the circuit) there, either. What to do?
The answer is, we insert 1 ohm resistors under the cathodes. We measure volts across those 1 ohm resistors. Every millivolt dropped across those resistors is 1 mil through the tube by ohm's law. The tube has no idea the 1 ohm resistor is there, 1 ohm is less than nothing to a tube. This allows us to stick 2 probes in there and measure/compute tube current and pull our probes away and we are done.
So just to make things entirely and bizarrely clear:
We do not care about "bias current"
We do not especially care about "bias voltage" (though, with experience, we know that we'd like to see about -38 to -45 volts on a pair of 6L6 output tubes, and if we had no meter and no 1 ohm resistors, we'd have to rely on that kind of inaccurate method of measuring bias)
We DO care about TUBE CURRENT or PLATE CURRENT but have a hard time measuring it.
Until we place those 1 ohm R's under the cathodes.
Then we use Voltage dropped across those resistors to figure tube current.