No cap at the rectifier? That 1K is just dead-loss and a slight reduction in buzz; you could get much less buzz with a cap at the rectifier so the 250V point is C-R-C-R-C filtering.
Hey, you won't even get 250V that way. Your working load is 250V/20mA or 12.5K, so 1K in front of the first cap spoils the peak-detector action. I get 209V at "250V". With 100u at rectifier, then 1K+499 and load, I get 244V, which is perhaps what you expected.
No reason why LM317 can't be floated at 75V or 750V. In ANY case it can't be bolted to grounded chassis, and any reliable insulator will stand 75V.
I do wonder if a cathode follower needs Regulated heat. If it is just a matter of using an available 12VAC winding, many tubes have 12V connections or kin.
I wonder if you can get 6.3V DC with a 6.3VAC winding and a FWB. It will be close.
Simple one-cap filtering, even at 15,000+15,000, gives non-zero ripple and especially significant higher harmonics. I'd almost rather have mellow 60Hz hum than a slowly declining series of 120, 240, 480 etc Hz harmonics running all around the chassis.
In general: 6.3V is tough when rectifiers have nominal 0.6V drop, 1V at peak, and you have two diodes in series. Significant rectifier loss, also high peak cap current.
Put your DC heaters in series. You got, what? Four 6DJ8? 24V 0.365A of DC is easier than 6.3V 1.4A of DC. The fact the "top" heater is up to 24V above the bottom one is unimportant.
The 75V bias on cathode follower heaters is surely not for hum-leakage, so I assume it is heater-cathode voltage rating. I am wary of the scheme: the bias comes to over 75V at switch-on, but that cathode will be stuck at ground until the tube heats and draws current 9 seconds later. If breakdown is really an issue, the heater supply should follow the cathode, not a fraction of B+.
Gimmic your cathode followers to draw 12V, add a shunt to pull 0.365A. Wire ALL your heaters in series, 36V 0.365A. This could even be 24VAC: gives 35V DC near-enuff. Put the cathode followers on top (+ end of the string). Shunt the preamp heaters with another few-thousand uFd... this uses the less-sensitive CF heaters as the R in an RC-filter to the more-sensitive preamp heaters. Where are your CF cathodes, around +120V? And preamp cathodes near 1V. Split the difference. Center of heater-string at +60V, or rather bias the bottom of the heater-string to +48V with only a few mA of available current. Put 0.1uFd from the heater bias to ground so the AC leakage is bypassed. To be really clever, take heater bias as a fraction of CF cathode voltage: zero when cold, rising as cathodes rise.
I think that is cheaper, more efficient, and even easier to wire. (Drawback: if one heater fails they all fail. But well-fed heaters rarely fail, and in a domestic stereo phono usually a one-side failure is a "total" failure anyway.)
FWIW: PCC88/7DJ8 is 7V 0.3A, can be seriesed with 12AU7-like 0.3A heaters, is fairly available and mostly "vintage" unlike the widely re-made re-marked 6DJ8. 25L6 is also 0.3A, cheap and old, and would make an authoritative cathode-follower, albeit at much higher total heat.
