Lobsterman claws can definitely kill you, especially if they hit you from the sides. I have tons of video evidence of this occurring in my latest playthrough. It's just not a guaranteed kill like the thermic lance.
You are most likely immune from the front though. I know displacers are practically immune to lobsterman claw attacks.
It's actually a viable tactic in alien base assaults to move close enough to the lobsterman to try and trigger a melee attack instead of a dpl launch.
And yes, TFTD armor is stronger than UFO armor from the front and back, but unlike TFTD damage, if an attack hits, it's guaranteed to do some damage, so more armor is needed in general. TFTD has a much harder early game than UFO, but it balances that out with a, I feel, more balanced end-game.