I never flash either. In case I might come across the need of darkening
highlights to make them show more detail, I use filter #00 of my VC head for
a _local_ burn-in. I occasionally make prints this way, none ever showed any
tracks of this kind of burning in, as it only affects the highlights and
leaves midtones and shadows unchanged.

You should try it then!

It needs to be carefully controlled, of course - which means you need a
second enlarger.  Stop down the lens to its minimum aperture; crank the
enlarger to full height then do a test strip with half the paper
covered throughout the process (use, say, 1 second intervals).  Count
the number of exposures!  This is important.

Now, when you look at the test strip, you need to pick the point before
the paper fogs.  This is where the covered half of the paper is
critical - it gives you a white to compare the other side of the paper
to.  Work out the point just before the paper fogs.  (for which you'll
need to know the number of exposures)

Flashing is extremely useful for blown highlights - especially where
burning isn't practical (e.g the highlights are caught up with other
detail)

The theory is that you are reducing the amount of light from the
negative needed to register on the paper - useful for very dense parts
of the negative.

You can also partially flash a sheet of paper - i.e one area only - to
avoid closing shadows & midtones in the other areas.

I should also say that I've been taught flashing by one the best fine
art printers in Australia.

But back to my question, Ilford Warmtone FB doesn't seem to respond
well to it - at least when I flash to a point just before the paper
fogs.  Maybe I need to back it off a little.