The signal levels are far smaller in preamps so we provide some shielding for heater wiring (sometimes, not always!) in those circuits. When you have 0.002V coming into something like a phono preamp, then every little bit helps. On the other hand, the Crack uses unshielded heater wiring and is adequately quiet (though it is lower gain and accepting much higher signal levels).
The signal level on the plate of the 300B can be hundreds of volts, so what could possibly get picked up by the grid is very tiny in comparison, especially considering the noise contribution of the filament inside the 300B is not insignificant.
As a test, you could float the shields on the heater wiring going to the 5670s. That shielded twisted pair wiring is there because it's an appropriate size for the current demand of a 5670 and it's good practice to use shielded twisted pair wires for AC heaters, but I would bet you'd hear no difference in noise with the shield floating.