Is this really true? I cannot mount an unapproved CD player someplace
so that it won't flop around in flight?

yes

Not true in the slightest. You want to velcro your CD player to the top
of the glareshield or maybe to the floor have at it. I have my
satellite radio attached to the far left side of the panel with that two
sided 3M sticky tape they give you with the radio. That stuff is better
than screws because when I want to remove it you'll never know it was
there and if you've worked with that tape before you know how solid it
is. I run the power, antenna and audio out cables under the glareshield
and then out to where they need to go.