Flipping it over a towel works until it doesn't. The trick only really helps with loose screws that are already rattling around in there, but the ones still threaded into plastic posts or deep in the body won't just fall out when you turn it upside down. You'll shake it, hear one drop, then spend ten minutes fishing out the last two with tweezers anyway. The magnetic mat is actually doing more than you give it credit for, since it keeps them from rolling into the carpet or getting lost under your knee. I still keep a mat around for that reason alone, even if I do the upside down thing too.
To be fair, the last phone I cracked open had 11 screws and 8 of them were still threaded in. Flipping it just dumped the 3 that were already loose and left me poking at the rest with a spudger. The mat earns its spot on the bench for catching the ones that actually fall, that part I'll give you.