Why do routers need to be reset?

Beside obvious reason of existing bugs in software there is a thing that we are dealing on hardware level such as changes in the memory due to cosmic radiation and noise (popular problem in high altitude observatories for example) and since int can change any memory including working registers, adress pointers, etc. it means that gradually memory in your device turns into noise. Professional devices often have a watch-dogs basically a circuit that have to be reset. In some cases when there is extreme availability is required such as fire alarms on petrol and chemical facilities or process control boards two spare modules constantly monitor each other and reset if needed. There is also a parity bits, ECC and lot of other options.

Interesting, thank you.

The problem is, I have clients who aren't about to plug/unplug power cords. I have set up a few with wireless appliance switches. Even that is far from elegant imo.

Because they're not willing to pay for reliable firmware.

The FVG318, while inexpensive, is supposed to be "business class."

The really reliable stuff is called enterprise-class or carrier-class.