G1 AC signal "just" gives the tube something to amplify (input sig +/- trem signal) not alter "bias"
The AC signal on any amplifier grid absolutely does vary the tube bias. That's how amplifiers work. Small ac voltage on the grid varies the bias and causes current through the tube (and plate resistor) to vary. This varying current through the plate resistor causes a larger varying voltage to be dropped across the resistor. Abracadabra. You have gain.
Let's do it differently... No signal, just a linear pot on the grid of a cathode biased tube. (Just for shits and grins let's call the pot INT.) We have a +12v on the top side of the pot and we have a -12V on the bottom of the pot. With the pot set to the middle we send zero volts to the grid. Let's call this the quiesent state. Zero volts causes some current to flow through the tube. A certain voltage will be dropped across the cathode resistor and a voltage will be dropped across the plate resistor. Everything just sits like that in a quiescent state. Now crank the pot all the way down to -12V. This will cause the tube to conduct much less, the cathode voltage will decrease and the plate voltage will increase. Notice that the plate voltage did just the opposite of the grid voltage. It's inverted. Now crank the pot all the way up to +12V. The tube conducts heavily. Cathode voltage goes up. Plate voltage goes down. Inverted again.
If you're still awake, try this... Crank that pot back and forth about 5 times per second. What do you think happens? Are you varying the bias? Could you call this bias vary tremolo?

It makes no difference if you vary the voltage on the cathode or the grid. The results are the same because bias is defined as voltage difference ***BETWEEN*** grid and cathode.