I agree. With dc, one end will always be at a high positive dc voltage with respect to the other end, and there will definitely be current flow across the vacuum.
This has all sorts of implications. The anode end would probably run considerably hotter, and then there is the problem ion migration.
In a thermionic diode the directly heated cathode is going to run at low voltage (typically 4 to 10v) and high current, and be a fairly massive structure compared to the skinny filament in a high voltage lamp. The anode would be even more robust.
Another probably more serious problem is what happens when the lamp finally gives up. With ac, the dreaded "plink" and sudden extremely bright flash at cold turn on, is actually an arc. The momentary half cycle fault current can be very high indeed, but at least with ac it will most likely be extinguished at the next zero crossing, (provided there is minimal series inductance).
With dc, a continuous arc might form which could possibly grow to frightening proportions at a high dc voltage with minimal series impedance, until the main fuse/breaker finally trips.
Interrupting large dc currents at high voltage is a very serious business indeed, as the fuse and circuit breaker manufacturers are well aware. What happens if a massive continuous dc arc escapes from inside the glass envelope ?