The explanations make sense to me.

The average *voltage* of an amplitude modulated signal equals the
carrier voltage, provided that the modulation has no DC offset (that is,
its average value is zero). This would be true for any waveform,
provided that it's AC coupled and it's not DC level shifted, clipped, or
otherwise distorted after the AC coupling. (It wouldn't be true of a
classically overmodulated carrier, for example.) The average of an AC
coupled waveform is always zero, if averaged over a time period that's
long compared to the time constant of the coupling network.

So a power meter that's really reading the voltage should stay at the
carrier level, provided that its time constant is comparable to or
longer than the time constant of the AC coupling of the modulation
signal. I'd expect this to be the case for a typical meter and typical
audio modulation.

Even if you're monitoring the true power level, the average typically
wouldn't get much greater than the carrier if you were modulating it
with a normal voice. Compression would tend to raise the average some,
though.

As I understand the Bird wattmeter, it basically takes current and
voltage samples and adds them (rather than vectorially multiplying them,
as a true power detector would have to do). What this would do to its
response to average power I'm not sure, but I wouldn't expect it to give
an accurate indication of either the carrier power or total power of a
modulated signal.

We're confronted all the time with measurements that seem to contradict
established theory. Some people regretfully are quick to embrace these
as evidence that established theory is wrong. It does take some effort
and knowledge to dig a bit to find out why there's a disagreement. But
with a miniscule frequency of exceptions, the digging always reveals
that we're not measuring what we think we are, we're using the wrong
theory, or we're applying it wrong.

Roy Lewallen, W7EL