Can electrons not flow through a load if they do not have a path to flow to the positive terminal of the battery? Why is it absolutely necessary for electrons to end up at the positive terminal?
It is not that it is just necessary. Electron flow, in the first place, OCCURS due to the potential difference between the 2 terminals - negative and positive terminals.
Emf or potential difference is the driving force of the electrons in the electric circuit which causes the electrons to flow from the negative terminal to the positive terminal.
If the circuit is not closed, electrons will not flow.
EDIT: As per the comment by brucesmitherson ;
In an open circuit, the driving force is not absent, it is just that it is not strong enough to make electrons jump outside of the metal, which requires a lot of energy.