semiconductors - Understanding reverse biased PN junction

I'm trying to understand a reverse biased PN junction, conceptually. For reference, I'll adopt the convention that the P-type is on the left, and N-type is on the right. So without a bias, the depletion region forms and there should be a positive electric-potential step crossing from P to N. This potential step deters more holes from diffusing into the N side, and electrons from diffusing into the P side. I understand that applying a forward bias will reduce this barrier, allowing carriers to cross the junction quite easily.

So applying a reverse bias increases the potential step, making it even harder for holes to migrate to the right, and for holes to migrate to the left. But here's where my confusion comes in: who's asking them to? If the positive side of the voltage source is applied to the N-side, shouldn't the electron current be pointed left to right, and the hole-current be pointed right to left? In that case, the potential "step" should be a "downhill", which the carriers should be happy to cross.

So what am I missing here?

Edit

I think I figured it out, can anyone confirm that the following understanding is correct? The N-side could, in theory, provide the electron current, and the P-side could provide the hole current. but to complete the circuit, a electron current will need to flow out of the voltage source's anode, and all the way through the P-side. Electrons entering the P-side should be able to fill in holes in acceptors, and migrate towards the right as holes drift left. However, when the electrons get to the depletion region, all (or most) of the acceptors have already been filled by the electrons which diffused over from the N-side. So not only are there few (if any) holes left to be filled in the depletion region, but there is also a net negative charge there due to the acceptor ions, so there will be a force pushing back on the electrons anyway. I guess if any electrons manage to get past this wall of acceptor ions, then they would get to slide down the potential hill towards the n-side, but the issue is getting that far in the first place.