Why can't you connect the + of a device to one battery and the - to another battery (and have it work)?
What you have drawn isn't a circuit, i.e., there is no closed path for an electric current.
Assume that there is a steady electric current through the bulb. Then, electric charge must be leaving one battery and arriving at the other - one battery is steadily becoming negatively charged and the other is steadily becoming positively charged.
But the increasing electric field from such an arrangement would act to reduce the electric current until it was zero.
Thus, the original assumption that there is a steady electric current is false.