Christine, I think we are in agreement, here.  The whole idea of playing
speed chess, is that the flags are going to be visible to those playing
the game.

It is exceedingly bad form to cover up the dial with one's large, lazy
fingers hanging over the side of the clock so you can't even see how many
minutes are remaining.  And that goes double with covering up the part of
the dial where the 'flag' rests.  I can remember thirty years ago, there
was this one guy who had the nerve to do this, and not only that, grope
about for the purpose of manipulating the settings on the reverse of the
clock, causing one player's clock to tick faster than the other's.  At
least in the olden days, independent and neutral onlookers could be
deputized by the tournament director with all the authority necessary to
call 'flag' and end the game, once the flag had fallen.  (But ask
yourself, seriously, how would this be possible if one or both players
took to draping their huge clumsy fingers over the dial, and obscuring the
flag, let alone poising them over the buttons that need to be clicked?) 
Then, when speed chess tournaments became more popular, one of the
players, by himself, had to notice the falling of the flag, and he was the
one that had to call it out.  (I wonder if this practice varied from place
to place?)

If FIDE actually permits alternation of the hands between moving the
pieces and clicking the button, there is probably a requirement that there
be a palpable period of grace between the moment the hand approaches the
clock, and the other hand rising to tap the button.