I can't tell you the thinking of organ factories, but from a
woodworker's perspective, you use plywood because it has dimensional
stability, moving about the same amount in each direction, and much
less than the cross-grain movement with humidity changes of a plain
sawn board.  If (and only if) similarly finished on each side, it will
also tend to remain flatter.  Sawn boards, even if quarter-sawn,
inevitably cup with humidity changes.

I do not find high quality Baltic birch, appleply, or similar hardwood
plywood to be cheaper than, for example, plain maple; so I don't think
cost was a motivator.

With regard to thickness, what do you need?  Given the range of
thicknesses available, it should be possible to come within a
millimeter of the previous thickness by laminating sheets of different
thicknesses together.  When I've needed thicker material than the 3/4"
or 1" plywood typically available, I've also laminated two pieces
together, then just run the assembly through the planer to get the
exact thickness I need.  One side will be ugly, because chances are you
will hit the glue lines.  But it can either be used on the hidden side
or covered with veneer, if you need a show face on both sides.

You could also plane one sheet and then put the planed surface on the
interior of your lamination.  That might take some trial and error if
you're trying to hit a really precise measurement. For bellows use,
it's hard to imagine that a millimeter or two in either direction will
make a huge difference.

Roger Wiegand
www.carouselorgan.com
<www.facebook.com/carousel.organ>