Correct me if I am wrong, but I think that typical batteries have a relatively high output impedance, certainly not less than one ohm output impedance broadband of our current active high voltage analog power supply designs.
Not sure what you mean by "typical batteries" but as far as SLAs are concerned, while I haven't seen any broadband measurements, at 1kHz (which is the frequency at which manufacturers rate internal impedance), it's quite low, though it depends on the battery's capacity. Larger capacity batteries will have a lower internal impedance than smaller capacity batteries.
For example, a Panasonic 12 volt, 1.3Ah battery has a rated internal impedance of 90 milliohms, and a 12 volt, 33Ah battery has a rated internal impedance of just 7 milliohms.
If you want to be assured of broadband, low impedance, you can always add some reservoir/bypass capacitance.
Battery power, fed into the typical tired old pi-network passive supplies, is not going to solve a thing.
I dunno. I guess it depends on what one considers a problem.
Personally I consider the AC line, the power transformer, the bridge rectifier and the noise they produce along with 120 Hz ripple on the "DC" side to be a problem. An unnecessary evil if you will. I'd prefer they not be there in the first place than having to try and deal with it.
Also, I am kind of wondering how battery power is going to cleanly supply isolated multiple near 300V DC supplies needed for proper vacuum tube operation.
Near as I can tell, your isolation comes from the multiple active regulators you use, not from the main supply, in which case I don't see that isolation has anything particularly to do with whether the main supply is an AC supply or a battery supply.
As for near 300 volts, that can be done with batteries and some have done it, though admittedly it would be expensive and recharging would be something of a nightmare so there are good arguments against batteries for high voltages.
Battery powered heater supplies? Gee how long can you play with batteries when the tubes need 600 mA each for proper heater operation. I don't think the energizer bunny will be up to the task. 
Nah. A high capacity 6 volt SLA would let you run the filaments all day. 300 volts is a nightmare, but 6 volts is a piece of cake.
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