Really, the best solution is for more Americans to ask for Robertson or
Torx drive screws so that your hardware stores start carrying them.
I've been doing DIY work for well over 20 years now, and I don't even
keep any Phillips or slot screws in stock. Everything I have in stock
and use now is either Robertson or hex drive.
Once you guys start using Robertson screws, you'll never want to go back
to Phillips drive.
So far as the stripped Phillips head screw goes, I'd try jamming a small
slot screw driver into the drive diagonally and try turning it that way.
Drilling the head off and twisting out what's left with a pair of vice
grips would be my next move. I agree that trying to grind a slot in it
with a Dremel usually doesn't work. That's because often the slot isn't
in the middle, and when you twist with the slot screw driver, you only
end up breaking off the smaller part of the head.
If you have good access to the head, another thing to try would be to
grab onto the head with a pair of side cutters and try to twist the
screw out that way.