Why Does RAM Have to be Volatile?

The Answer

 it’s due to physics.
Any non-volatile memory must store its bits in two states which have a large energy barrier between them, or else the smallest influence would change the bit. But when writing to that memory, we must actively overcome that energy barrier.
Designer have quite some freedom in setting those energy barriers. Set it low 0 . 1, and you get memory which can be rewritten a lot without generating a lot of heat: fast and volatile. Set the energy barrier high 0 | 1 and the bits will stay put almost forever, or until you expend serious energy.
DRAM uses small capacitors which leak. Bigger capacitors would leak less, be less volatile, but take longer to charge.
Flash uses electrons which are shot at high voltage into an isolator. The energy barrier is so high that you can’t get them out in a controlled way; the only way is to clean out an entire block of bits.

In other words, the only way to make the RAM as high speed as we need it for modern computer operations is to keep the resistance between state changes extremely low (and thus make the RAM volatile and susceptible to data erasure in the face of power loss).