I don't think the unbalanced current draw matters. It's as if you had 2 separate filament trannies, ea capable of 6.3-0 @ 4A. Here, the secondaries "happen to" share the same primary coil, and are wound out of phase so that their 0's are conveniently located in the middle, creating 6.3-0-6.3 @ 4A. So the 2 secondaries don't know or care if they have balanced current draw. (Maybe the primary will begin to care at some point, but let's assume the designers did their job well enough.) Current in excess of 4A will drop voltage, but as long as 5V remain, the tubes will be happy.
My question: is it noisy? If the heaters are quiet and free from hum, then the stock circuit is OK. But in some old PA amps, some level of hum may have been tolerated. The one-sided ground scheme at ea tube negates the benefit of humbucking by using twisted pairs of filament wires. Not so much an issue with PP power tubes, because the PP arrangement is noise & humbucking anyway. It might be an issue with other tubes. OTOH, twisting filament wiring is considered over-rated by some; and many quality vintage amps did not do so with no ill affects.
If you need or want to use twisted pairs to buck hum, there's another alternative: Make a series-pair of 2X 6.3V tubes. This sums to 12.6V for ea series pair-- BUT the filament current draw must be the same for ea of the two tubes in a series pair. Wire the series-pairs in 2X parallel strings from the tranny. Use a pair of ground reference resistors for each string.
If there's an odd tube with no mate for a series pair it can be left one-side grounded. For lowest hum, do this with a tube late in the chain and/or with a low noise tube (not an EF-86 if possible). Any heater hum that originates early will be re-amplified in later gain stages.