On the surface this sounds like a real danger... if you put current through a transformer then suddenly remove the current source, inductive behavior will cause a voltage spike. We know this can be dangerous in some cases, that's why you never run a tube amp with no speaker (or resistor) attached!
I doubt as little as 1ma is enough to cause problems, though. Transformers have some internal capacitance and other imperfections that will soak up small amounts of energy before they can create dangerously-high voltage spikes.
I think my skepticism is backed up by the thousands of people who have used DMM resistance-meter settings to examine their transformers and NOT killed them.
Note the "real" danger here is a voltage spike which breaks through insulation, shorting stuff out inside the transformer. This won't cause "distortion" but instead, a seriously messed up transformer. Similar to what happens if you run an amp with no speaker attached. It's not subtle when it happens!
A lot of other stuff on that page is nonsense. Permanently damaging a transformer by "magnetizing" it? "Fixing" it by running AC current through it? The idea that it IS safe to test a transformer this way, if it can handle DC current? None of that is real.
A single-ended transformer (meant to handle DC) can be killed by inductive spikes, many of them have been killed by running without a speaker attached. So if this is a danger for a push-pull transformer (I'm skeptical) it is also a danger for one that can handle DC.
As far as I know, core saturation is an effect that goes away pretty much as soon as current is removed.
And lastly, if push-pull transformers truly couldn't handle ANY DC current, they'd be damaged every time your PP output tubes bias current is unbalanced by 1ma, or if your phase inverter sent an unbalanced signal. None of that happens. It will produce increased distortion WHILE those imbalances exist in the circuit, but not once the transformer is returned to proper use.