Why bank and rudder together produce zero sideslip

Left engine out: the right engine's thrust yaws the nose toward the dead engine. Right rudder stops the yaw.
But the rudder is a lifting surface. The side force it produces on the tail acts toward the dead engine, and with the wings level nothing opposes it - so the airplane slips through the air, fuselage skewed to the relative wind.
Bank ~2° toward the operating engine. The lift vector tilts, and its horizontal component acts toward the good engine, opposing the rudder's side force.
With the two in balance, the airplane no longer has to slip to make up the difference. The fuselage aligns with the relative wind: zero sideslip.

Why that leaves the ball off center

The horizontal component of lift is what normally turns an airplane. Here it is fully committed to opposing the rudder, so the net sideways force is zero - the flight path stays straight and the heading holds.
In a normal coordinated turn the ball centers because the bank produces a turn, and the turn's inertia pushes the ball outboard by exactly as much as gravity pulls it toward the low wing.
In zero sideslip there is no turn, so nothing offsets gravity. The ball responds to gravity alone and settles toward the low wing - one-third to one-half a ball width toward the operating engine. Its displacement is really just reading your bank angle.

Why this contradicts the single

In a single, a slip (ball toward the low wing) and a skid (ball toward the high wing) are the same thing in opposite directions - both are sideslip, and both show on the ball.
What a single doesn't have is a standing sideways force. The rudder can certainly make one - a forward slip is exactly that - but it is a force you are choosing to hold, and releasing it removes the force and the sideslip together. Centering the ball and zeroing the sideslip normally coincide, which is why "step on the ball" works in either direction.
With an engine out, the rudder you hold is opposing a yawing moment from thrust, not from the relative wind. It stays there whether or not the fuselage is aligned, and the ball - which only sees the sum of the forces - cannot tell the two sources apart.
So "balanced" now has two solutions: let the air balance the rudder, which takes a sideslip, or let the horizontal component of lift balance it, which does not. Only the second is zero sideslip.

In a single, a centered ball means no sideslip. In a twin with an engine inoperative, it means no bank - and with rudder held in, no bank means you are slipping. Chasing the ball back to center from a correct 2° of bank takes still more rudder, restores the sideslip, and turns you instead of holding heading.

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