Mech arms are perfectly precise and dextrous within their designed range of movement, multiple sources state this. They possibly even compensate for pilot clumsiness to some degree, if that was ever needed.
It is, however, rather obvious that not all mech arms are capable of a human arm's full range of movement. For example take the Awesome's left arm. It has a hand, but with that armour shield, it'll never be able to raise that arm up sideways, it's shoulder is basically an axle joint, not a ball joint (in the words of the Terminator: Why didn't god use more ball joints when he made humans?).
The ability to stow a weapon on its person(?) might simply not be an efficient design for mechs. The quickdraw with the lift hoist arrangement from one of the XTROs seems to require extra equipment to do so, it stands to reason most designs would.