That's actually pretty close, but one little step missing.
As you say, there is a lift component going 'sideways' for a banked aircraft. That isn't, however, helping you directly with 'anti-yaw'. What has to then happen is that in order to oppose the sideways acting lift component, you need to generate an opposing force; that you can do by adding a little bit of sideslip. That sideslip not only generates the balancing sideforce you need for the bank, it also generates a yawing moment which assists you in opposing the yawing effect of the dead engine.
The lighter the aircraft is, the less the size of the sideways lift component, the smaller the amount of sideslip you need, and the smaller therefore the assistance the sideslip gives in yaw control.