> need maybe 6 amps of filament current.
No.
You ask if 6V AC near circuits can add hum. You have 0.020V signal at first tube, 40V signal at final tube. Which one will be most affected?
Generally there are three hum-critical stages: first stage, stage after volume pot, and reverb recovery. See Alerich's Marshall plan: only V1 preamp and post volume/tone network gets DC, the driver is AC heat and the EL34 sure are AC heat.
Generally in hand-wired, you can twist and route heater lines well away from audio paths and get hum way far down.
Often on PCB amps, when budget won't allow flying heater lines, DC heat is "necessary"; but only on the little tubes, not the big ones.
> That implies a fair sized heat sink
No. You need clean DC (rippled DC is worse than pure AC), but you do not need Regulation on DC any more than on AC.
In general, the ~~1V drop per rectifier adds-up on 6V DC circuits; that's why Peavey generally wires their small tube heaters in series at ~~25V DC. 2V drop on 6V useful is 25% waste; 2V drop on 25V useful is 8% waste. (It also makes less traces on the PCB.)
> artificial ground for the DC part but not for the AC part.
The AC part is effectively centered around ground via the rectifier and DC CT. (If the DC were any good, it would not need a CT.... either lazy thinking or some empiracal design here.)