This answer is based largely on a critique of this video:
The video is interesting but I think it contained some fundamental errors in theory if not result. From 8:54-9:12 the author is speaking as if sideslip is a result of side force. That's essentially an Aristotelian concept rather than a Newtonian one. Sideslip is not a result of sideforce-- turning is. Rather, sideslip is a result of not being pointed the same direction you are actually going. Therefore it's inaccurate to say that when you are banked, the weight vector has a component that causes sideslip in one direction, or opposes sideslip in the other direction.
(Btw this pertains to related discussion in the answer
Could a plane be constructed to be fly in fixed-stick roll-stable circles? -- specifically the comment "From the point of view of the aircraft, lift is still acting in the plane of symmetry, but gravity does not and will cause it to sideslip.")
What's really going on in the twin-engine case is that the slip-skid ball reacts to all aerodynamic sideforce components (but NOT to the sideforce component contributed by gravity, because gravity accelerates the aircraft and ball together-- this can also be explained in another way involving "centrifugal force".) When one engine fails on a twin-engine aircraft, if you want to exactly center an imaginary yaw string at the nose for maximum streamlining, the ball cannot be fully centered, because the rudder is strongly deflected and creating some sideforce toward the dead engine. That's why you leave the ball deflected about half a width toward the good engine--because it streamlines the fuselage. And fundamentally the rudder, not the bank anlge, is the control what controls the position of the slip-skid ball.
The purpose of the banking toward the good engine is to stop the turn that would otherwise result (due to the rudder sideforce) when the ball is in the optimum position, NOT to influence the amount of sideslip that is present. Similarly it is incorrect to suggest the banking TOO FAR toward the good engine would slip the plane sideways through the air toward the GOOD engine and possibly stall the vertical tail as a result, as the author suggests from 9:25 through 9:28.
Now, granted, if we are simply using the ailerons as needed to establish and maintain a set bank angle, and using the rudder as needed to hold heading while flying at that bank angle, then FUNCTIONALLY things would end up working much as the author says. (Except that the possibility of stalling the tail due to "too much bank" seems far-fetched-- are we really contending that the pilot is applying so much rudder toward the good engine that a yaw string would be deflected extremely far toward the good engine, despite the thrust imbalance? I don't think this is the true purpose of the 5-degree bank limit cited in the video.) At any rate, if we are flying in this manner, attempting to control heading with rudder, then too much bank could indeed produce some slip toward the GOOD engine, and too little bank would produce some slip toward the bad engine, both as indicated by a yaw string on the nose (rather than by the slip-skid ball-- though they will both basically agree whenever we are talking about huge deflections.) And when you lose an engine, initially one your first concerns probably is to apply rudder as needed to minimize yaw rate, so you might end up applying the controls in just this way. But fundamentally, the bank angle is not causing or preventing sideslip. Rather, the rudder is -- along with the yaw torque from the one working engine. In the long run, the bank angle is controlling turn rate, not sideslip.
IF airspeed is sufficient to give sufficient rudder authority to maintain the correct position of the slip-ball-- displaced about half a ball-width toward the good engine, then there ought not be any problem with turns in either direction.
On the other hand if you are fighting for basic control over the aircraft and you are having trouble preventing the aircraft from yawing and rolling toward the bad engine even with a lot of rudder applied-- which likely means the ball is still displaced rather far toward the good engine-- then the last thing you would want to do is bank toward the bad engine. The additional turning tendency would exacerbate the difference in airspeed between the two wingtips and cause the aircraft to tend to roll toward the bad engine. Banking toward the good engine will have the opposite effect and help you prevent the aircraft from rolling toward the bad engine.