GaryB wrote:
Grover,
When you connect the transformer to the output tabs of the bridge rectifier, you have 2 diodes in series across the transformer. On one half of the AC cycle, these diodes are reverse biased and effectively out of the circuit. On the other half of the cycle, the two diodes clip the voltage at ~ 2 diode drops or ~ 1.2 volts. A 5V AC transformer has peak to peak voltage of 5v x 1.414 = 7.07v or referencing things to ground, the AC voltage will swing +- 3.535v. So on one half of cycle you'll be clipping the AC, which is why you're measuring the reduced voltage.
---Gary
...and that's exactly what I measured, Gary. Thank you for explaining it!

Regarding the DC arrangement, I did add a filter cap to the output of the bridge. I'd need 6.3V to add a common-mode choke or extra RC filter. It was a quick-and-dirty experiment.

It sounded pretty good but would need refinement.
When I built an SE 845 amp, I definitely needed DC on the filaments and used a CM choke to smooth it out. The sound was pretty good. I might pursue this with the 300Bs.