Mewtwo is blessed with an excellent move pool that has tremendous fire power and excellent coverage, firing off of a massive 154 Special Attack. And with STAB Psystrike firing off an essentially Base Power 150 Psychic type nuke, you don't really need to bother about pressing other buttons (Dark, Steel, other Psychic Pokemon being the only exceptions).
But that's not all. Say you do use Me First. The only times you're better off using that is when the opponent's attack is super effective against itself, limiting it to Dragon and Ghost type Pokemon for the most part. And most Pokemon resist their own type, so the 1.5x buff will simply be neutralized to 1x due to resistances, so you're better off using your own move anyway. By this I mean that using Me First to counter a Darkrai's Dark Pulse rather than Focus Blast/ Aura Sphere might backfire upon you and Mewtwo will faint.
Simply having Psystrike/Psychic and any of Ice Beam / Thunderbolt / Aura Sphere (or Focus Blast) / Shadow Ball / Flamethrower (or Fire Blast) puts you at a greater advantage.
I'm not saying Me First isn't a very good move, but it is wasted on Mewtwo which has much better moves to use in its limited 4 move slots. It is at best a gimmick that might get you a clutch win or two, but the sheer statistics will prove that consistency of damage is better in the vast majority of cases.